Ahmed Ben

Salah Ab

Nicolas Ragot
email: nicolas.ragot@univ-tours.fr

Thierry Paquet
email: thierry.paquet@univ-rouen.fr

Salah

Adaptive Detection of Missed Text Areas in OCR Outputs: Application to the Automatic Assessment of OCR quality in Mass Digitization projects

Keywords: OCR results assessment, Text detection, Texture characterization

The French National Library (BnF * ) has launched many mass digitization projects in order to give access to its collection. The indexation of digital documents on Gallica (digital library of the BnF) is done through their textual content obtained thanks to service providers that use Optical Character Recognition softwares (OCR). OCR softwares have become increasingly complex systems composed of several subsystems dedicated to the analysis and the recognition of the elements in a page. However, the reliability of these systems is always an issue at stake. Indeed, in some cases, we can find errors in OCR outputs that occur because of an accumulation of several errors at different levels in the OCR process. One of the frequent errors in OCR outputs is the missed text components. The presence of such errors may lead to severe defects in digital libraries. In this paper, we investigate the detection of missed text components to control the OCR results from the collections of the French National Library. Our verification approach uses local information inside the pages based on Radon transform descriptors and Local Binary Patterns descriptors (LBP) coupled with OCR results to control their consistency. The experimental results show that our method detects 84.15% of the missed textual components, by comparing the OCR ALTO files outputs (produced by the service providers) to the images of the document.

INTRODUCTION

The collections of the French National Library (BnF) contain works of art and documents that have marked the history of the humanity. For preservation purpose most of these documents are not easily accessible. To present these documents to the public, the French National Library has launched its first mass digitization program in 1996. Gallica † , BnF's digital library, contains the images produced since then. Before 2006, document queries on Gallica were performed using bibliographic references of the documents. Since 2006, to allow full text search, the BnF has launched mass digitization projects including textual transcriptions. This is a big issue since the document collections from the National Library of France are very variable in terms of physical characteristics. Indeed, they contain documents that come from several periods of printing.

The transcriptions are carried out by OCR providers that deliver ALTO 1 files. These files include the word transcriptions with associated word confidence computed by the OCR and the segmentation results. In mass digitization projects, the BnF ask the OCR providers either Raw Quality or High Quality (HQ) transcriptions. Raw Quality is directly obtained by the OCR output when the estimated word recognition rate, computed using the word confidence scores, is between 60% and 99%. High Quality requires an estimated word recognition rate of 99.99%. Of course, current technologies cannot provide High Quality outputs by themselves, and most of the time, manual corrections are required. It has been estimated that below an estimated word recognition rate of 85%, manual corrections would be both tedious and costly. Therefore manual corrections are asked only for documents that exhibit OCR outputs with an estimated recognition rate higher than 85%. Of course, the reliability of commercial OCR systems depends highly on the physical characteristics of the document. Moreover, the quality of the transcription can only be estimated by the system that provides a word confidence ratio for each recognized word. This is why an OCR verification stage should be introduced at the BnF before integrating the digitized documents into the Gallica digital library. Full human quality control would be too long when performed on the whole digitized corpora (around 30000 pages a day). Consequently, automating this control or part of it can be very useful.

The BnF digitization department has identified that most of OCR errors are coming from a significant amount of missed text components in the OCR outputs. This is why this study focuses on the detection of these missed textual components. Because of the complexity of OCRs architecture, 2 it is rather difficult to identify the reasons of these defects. Sometimes they occur at the pre-processing stage (such as binarization). Sometimes, defects come from the segmentation process, while some other defects come from the character recognition stage. The approach we propose to detect the missing textual components does not concentrate on the defects of each of the processing stages. It provides a single methodology able to detect the missed textual components such as characters, words or text blocs. In fact, our approach exploits the local information inside a given page and the ALTO result produced by the OCR provider in order to detect the missed textual components. More precisely, textual areas detected by the OCR are used as training samples for a classifier whose task is to discriminate between text and other elements in the document image. Such page dependent procedure provides a generic methodology able to perform on any kind of document and particularly on old documents for which standard OCRs perform moderately well and require a lot of verification efforts at the BnF.

In section two of this paper, we analyze the various situations of missed textual elements. We try to explain the reasons of these defects by referring to the complexity of the OCR processing chain and we explain why these defects sometimes escape from quality control. In the third part, we describe the methodology of the proposed approach. Next, we report the experiments that have been carried out on a set of document images coming from the BnF collections. Then we conclude this article with some possible improvements of this work.

OVERVIEW OF OCR DEFECTS

An OCR system is composed of multiple processing stages organized sequentially. It includes preprocessing and binarization of the image, segmentation, and character and symbol recognition. This sequential organization tends to cumulate (even amplify) failures done at each stage. Nagy et al [START_REF] Rice | Optical Character Recognition: An Illustrated Guide to the Frontier[END_REF] show that preprocessing and segmentation failures can result from physical defects of document such as ink defects and paper defects. Furthermore, document image defects such as low contrast and background noise may lead to wrong segmentation results. Regarding the character recognition stage, errors may come from exotic character styles or may be due to the presence of noise. These errors cannot be detected easily by analyzing the OCR output only, since they are forgotten during the OCR process either during preprocessing and segmentation stage or during the recognition process when they induce a low word confidence.

According to the BnF digitization department, most of the errors in OCR outputs are missed textual components. The examples presented in figure 1 show the diversity of the cases that can be encountered. In this figure, the red and green areas refer respectively to text and illustration elements detected by the OCR. Figure 1 (I) shows a large missed text block. It is rather difficult to explain these failures, whereas some nearly similar components are correctly detected in the same document. Sometimes one single word is missed in a well segmented block (cf. figure 1 (III)). In this case, we may assume that text in italic has low word confidence scores and may be rejected at the end of the character recognition stage. Also in some cases, as in figure 2, one single character is missed probably due to its exotic shape.Figure 1 (II) also shows some missed textual components in the OCR output. These missed elements might occur because of different phenomena. For example, the presence of noise (bleed through effect) may disturb the character recognition stage. Some text areas are also missed in the title, although the character shapes do not exhibit any difficulty for the recognition stage. Finally, we may conjecture that the italic style has also disturbed the recognition stage. One can also notice that the illustration (I) (II) (III) Figure 1. The labeled mask areas : (I) case of a missing section in the OCR output, (II) case of a missing words, sentences and graphical elements in the OCR output, (III) case of missing words in the OCR output. The red bounding boxes represent the recognized words by the OCR, the green bounding boxes represent the detected graphical elements by the OCR in the upper part of the image is missed although it should be easily detectable since it does not overlap with any other components. The reason for this error is not clear.

Most of the OCR systems try to assess the quality of the output delivered. Most of the time, it is done at the final step, estimating the performance of the character recognition stage by the computation of a word confidence score. This one is obtained by combining the confidence of the character recognizer with the word matching score to a dictionary. This word confidence score ranges from 0 to 1 but its definition depends on the OCR provider and on the OCR technology. Based on it, the page quality score is the average word confidence over a page after rejection of the poorly recognized textual elements (those with a low word confidence). The maximization of the page quality score can then be obtained either by maximizing the recognition rate or simply by increasing the number of rejected text components. As a consequence, documents with a high page quality score may have missed textual components. Thus, if one look only at the page quality score to assess the quality of the digitization process it may leads to inconsistencies. This is why, in order to control the quality assessment of the output of a mass digitization process, it is necessary to design a post processing stage able to quantify the OCR defects in number of missed textual components.

DETECTION OF MISSING TEXT COMPONENTS IN OCR RESULTS

Overview of the method and state of the art

Within this study, our aim is not to improve a segmentation process by providing feedbacks to an OCR or to the OCR provider but to give to the BnF new quality insigths about the documents by detecting textual components (blocks, lines, words, and characters) that the OCR was not able to find. Let us call them the missed components. Compared to the huge amount of works dedicated to document image segmentation, this study investigates new incites of page layout segmentation oriented towards document segmentation verification. This subject of detection of missed components in OCR outputs has not been much considered until now, but appears to be of primary importance for managing digitization workflows at the BnF.

Considering that the BnF's collection of documents is very heterogeneous -since it covers several centuries of printed documents -we must investigate a general method to detect the missed components, that does not depend neither on the typography nor on the style of documents. We assume that the missed components can only occur in empty areas or background e.g. outside the word bounding boxes and outside the graphical regions already detected ‡ . The idea of our approach is to learn the properties of each element detected by an OCR and to try to find similar elements in the empty regions. In this way our method fits very well with typographic and structural characteristics of the current document to detect missed components inside it.

According to the state of the art presented by Chen et al, [START_REF] Chen | Text detection and recognition in images and video frames[END_REF] low levels features such as color, texture and shape characteristics are usually used to extract textual elements. Zhong et al 5 assumes that spatial characteristics of textual elements can be used for a connected component analysis to detect some text inside an image. Other works, such as the approach proposed by Smith et al 6 use the vertical edges of text with a predefined template to group edges and find the text elements in an image. Other approaches use a priori knowledge to locate the text elements. For example, Sobottka et al's 7 approach is based on specific text features, such as baseline feature, in order to identify text in images. In fact, strings are characterized by a top and bottom baseline that help to determine the presence of text in an image area. While this approach is applicable with serif fonts, it may not give good results for old documents that have exotics fonts. Garcia et al. [START_REF] Garcia | Text detection and segmentation in complex color images[END_REF] use the fact that text strings contain edges in many orientations to detect text components in an image. The proposed method combines the variation of edge orientation computed in local area with edge features to describe the text components. The main advantage of such model-driven methods and a priori methods is that they are very fast. However, they adapt hardly with heterogeneous documents collections. In fact, model-driven approaches are generally used for specific fields of application, while the BnF collection includes several kinds of documents (press, scientific books, maps, letters, Fax....) and several printing technology, this kind of approach is not suitable in our context.

A texture approach was proposed by Wu et al [START_REF] Wu | Finding text in images[END_REF] to detect and locate the text in an image. For each pixel, energy features are computed on derived images at different scales. The family of texture descriptors allows the characterization of document images with low-level information. This kind of features allows to locate text components without using any further information about the physical structure model or the typographical model of the documents. This characteristic corresponds to the diversity of the BnF's collection and this is why we used a texture approach to detect the missed components. According to the state of art presented by Journet et al, [START_REF] Journet | Document image characterization using a multiresolution analysis of the texture: application to old documents[END_REF] there are four types of texture descriptor: statistical descriptors, probabilistic descriptors, frequency descriptors and geometric descriptors.

Statistical methods such as Grey Level Co-occurrence matrix proposed by Haralick et al 11 characterize the regularity, the repetition and the contrast of textures. Other statistical approaches, such as the work of Uttama et al [START_REF] Uttama | Top-down segmentation of ancient graphical drop caps: lettrines[END_REF] use the autocorrelation function to describe the texture pattern. The statistical approaches study the grayscale intensities of pixels to compute the texture feature. According to Mryka, [START_REF] Beyer | Glcm texture : A tutorial[END_REF] the statistical methods have a good performance. However, they are not very suitable for old document images. In fact, the grayscale images of old documents are similar to binary images due to the printing technologies. Therefore, the variation of pixel intensity in the image may be caused by digitization process and not by texture. For this reason we cannot apply the statistical methods in our context. Caron et al [START_REF] Caron | Power law dependencies to detect regions of interest[END_REF] use an approach that belongs to the family of probabilistic methods to identify areas of interest in a image of natural scene. This approach use the power law (Zipf's law) to perform the detection of the textual areas.

Raju et al [START_REF] Raju | Text localization and extraction from complex color images[END_REF] propose another method for text detection from complex color images that belongs to the family of frequency methods. This approach performs text localization using a Gabor function based on multi-channel filtering of the intensity component of the image. But according to Journet et al, [START_REF] Journet | Document image characterization using a multiresolution analysis of the texture: application to old documents[END_REF] in some cases , eg. old documents, the frequency approaches for texture description may do some misclassification between the drawn illustrations and the text.

The geometric methods characterize the texture shapes and their spatial relationships. Tuceryan [START_REF] Tucceryan | Moment-based texture segmentation[END_REF] uses geometric moments to characterize textual texture in document image. Another method proposed by Journet et al [START_REF] Journet | Document image characterization using a multiresolution analysis of the texture: application to old documents[END_REF] for text/graphic separation is based on an autocorrelation function to study the main orientation of texture. In fact, in multi-resolution process, the texture of the text components has only one main direction at different scales. Whereas, the texture of graphic components has different main directions at different scales. The geometrical approaches are applicable both on binary images and grayscale images. Also, some geometric features as main direction of texture are suitable for the characterization of any kinds of fonts or graphics components (printed or drawn). This is why we have chosen to develop a new approach based on geometric techniques to detect the missed text components.

In our approach, the areas detected by the OCR provide a labeled mask of the image (cf. figure 1) which separates the background information from the foreground information (either graphics or textual elements). The foreground information is used for training a specific text and graphic detector assuming that the information provided by the OCR is reliable. Then the background information is analyzed using this detector in order to detect the missed textual components, if any. The proposed approach must deal with all kind of documents in the BnF's collection. This is why we need to train classifiers with the local characteristics of each page (both text and graphic elements). However, the characteristics of background and inter-word space are learned using a set of images § that includes binary images and grayscale images. Learning is performed by training four SVMs classifiers using the signature of the pixels of one given area as positive examples and others as counterexamples (one versus all classification scheme). The choice of the classifier was made after several experiments of supervised classifiers. The best results were obtained using SVM with linear kernel. After this learning step, pixels falling outside the initial labeled mask are classified as text, graphics, inter-word spaces and background. The inter-word spaces areas are then merged with empty areas in one single class corresponding to background areas. A final connected component analysis is conducted on text regions in order to provide an estimation of the number of missed textual components in the image. More details about the connected component analysis are given in the section 3.3.

Characterization of the areas

The proposed method was conceived so as to adapt to the various document contents that can be encountered with either textual or graphical content. Following this general goal, we choose to characterize regions with general purpose texture features that can be applied both on black and white and gray level images, since the BnF collections contain both kind of images. The multi-scale Radon features [START_REF] Deans | The Radon transform and some of its applications[END_REF] have been chosen for their ability to characterize oriented textures such as textual images. LBP features [START_REF] Ojala | A comparative study of texture measures with classification based on featured distributions[END_REF] have also been chosen as general purpose texture descriptors. Finally, a paper to ink transition descriptor and the mean pixel intensities have been introduced so as to characterize textual areas. So for each pixel we have extracted 12 features to characterize the textures of the image. The following subsections summarize these feature descriptors. § To characterize the background regions, we built a set of background images that includes binary images and grayscale images. This set of images contains some image defects (such as white noise) and some physical defects (such as small inkblot) .

Radon transforms descriptors

The Radon transform allows identifying the main direction of a texture. This information is very useful in our work. Indeed, according to Journet et al, [START_REF] Journet | Document image characterization using a multiresolution analysis of the texture: application to old documents[END_REF] printed areas should exhibit a constant orientation at different scales. On the contrary, the main direction of the texture should change at each scale on graphical areas. The behavior of Radon transform on background areas is similar to its behavior on graphical areas: various orientations at each scale. However, the intensities of its responses are weak on background areas compared to those on informative areas. The Radon transform computes projections of the image along some specified directions by rotating the image around its central point. According to Deans, [START_REF] Deans | The Radon transform and some of its applications[END_REF] the Radon transform of f (x, y) is the line integral of f parallel to the y'-axis (along y-axis).

R k (θ) = + l 2 -l 2 + l 2 -l 2 (f (x cos(θ) -y sin(θ), x sin(θ) + y cos(θ))d y d x (1)
In this study, we applied the Radon transform on three windows of size 128 × 128 (k = 1), 64 × 64 (k = 2) and 32 × 32 (k = 3) and seven directions θ ∈ {0, 30, 45, 60, 90, 120, 150} so as to identify the main orientations of the texture at each scale. Thus our Radon index δ k (i, j) is given by:

δ k (i, j) = arg max θ (max(R k (θ))) (2)
with θ ∈ {0, 30, 45, 60, 90, 120, 150} From the δ k (i, j) value, 7 texture descriptors (f 1 , ..., f 7 ) are computed. The first one measures the consensus of the principal direction at each scale. This means that for each pixel, we compute the principal direction of the R function at each of the three scales. If the principal direction is the same, our descriptor value is 2. If the main direction is the same only in two scales its value is 1. Otherwise, its value is 0.

f 1 (i, j) = (δ 1 (i, j) == δ 2 (i, j)) + (δ 2 (i, j) == δ 3 (i, j)) (3)
The next three features are the median over the maximum projection of each orientation computed at each of the three scales. By definition, the maximum intensities of orientations are large on well inked areas(illustrations), less important on printed areas and very low on background regions which are very bright most of the time.

f k+1 (i, j) = median(max(R k (θ))) (4)
with θ ∈ {0, 30, 45, 60, 90, 120, 150} and k = 1, 2, 3.

Similarly, the variance of maximum projection over each direction is computed at each scale. The variance is large on regions that exhibit several directions. It is low on textual areas.

f k+4 (i, j) = std(max(R k (θ))) (5)
with θ ∈ {0, 30, 45, 60, 90, 120, 150} δ k (i, j) and k = 1, 2, 3.

Local Binary Patterns descriptors

The original implementation of the LBP operator worked with the eight neighbors of a considered pixel. The sign of the difference between intensity of the central pixel and the intensity of neighboring pixels is used as a threshold to product a signature. This represents information on regular patterns in the image, in other words on the texture. Let (x c , y c ) a considered pixel with gray level g c and g p (p = 0, ..., P -1) be the gray values of the P neighboring pixels on the circle of radius R(R > 0). The LBP descriptors used here are given by equation 6. . This analyzes the regularity of the texture. Moreover, it is invariant to gray scale shifts since the signs of the differences of g p -g c are only considered.

f k+7 = LBP P,R (x c , y c ) = P -1 p=0 s(g p -g c )2 p (6) (I) (II) (III)

Paper and ink transition descriptor

We also introduced the characterization of textual areas by measuring its average paper / ink transition. The descriptor we designed is inspired from the works of Journet et al. [START_REF] Journet | Document image characterization using a multiresolution analysis of the texture: application to old documents[END_REF] Considering a sliding window of size 11 × 11 (the size was chosen empirically), we compute the average, over the window, of the line accumulations of horizontal differences between two adjacent pixels:

f 11 (i, j) = Avg j∈J ( i ∈I |p i,j -p i+1,j |) (7)
with I and J the size of the window and p i,j the gray level of the pixel of coordinate (i, j)

Mean intensity of pixels

Finally we present here the last feature that we have used in the signature of the texture. The pixel intensity is an important feature to separate background pixels from ink pixels. In fact, this characteristic is very reliable to separate the pixels that exist in the spaces between the words from ink pixels. The intensities of the pixels are very sensitive to image noise. To avoid this problem we used the average intensity of pixels instead of pixel intensity in the texture signature. The average intensities of pixels are measured on a window size of 5 × 5.

f 12 (i, j) = Avg j∈J ( i ∈I p i,j ) (8)
with I' and J' the size of the window and p i,j gray level of the pixel of coordinate (i, j)

Detection of missed text components

As mention in section 3.1, the classification of the pixels located in the background is performed by 4 SVMs classifiers to identify text, graphic, inter-word and background areas. After classification step, the inter-word spaces areas are then merged with background areas in one single class corresponding to background areas. Figure 3 shows the results of this classification step at pixel level. The red pixels are the pixels classified as text, the green pixels are pixels classified as graphics and blue pixels are background pixels. Black pixels are the pixels that belong to the labeled mask provided by the OCR.

Our quality verification approach should provide an estimation of the number of missed text components at word level in order to be suitable with the BnF's process of OCR quality control. In most cases, the missed text components detected by the classifiers represent one or two characters. This is why it is needed to merge the neighbouring detected text components to form words or pieces of word. This is performed using a distance transform and a dilatation process. The distance transform [START_REF] Bailey | An efficient euclidean distance transform[END_REF] measures the distance that separates the background pixels from text components detected in the image (cf. figure 5 (b)). We then group all the pixels surrounding a text component with this component, if the distance is less than δ (δ = 3) pixels to the component. The value of δ was chosen because it corresponds in most cases to the distance that separates two characters in a same word. This way, we obtain the envelopes of missed text components (called here Detected Missed Text Components DM T C). The proposed approach tends to make some detection errors on the examples when there are transparency defects or binarization defects (cf. figure 3 (II)). To eliminate noisy elements, we remove those that are smaller than the smaller text element detected by OCR in ALTO file. The number of detected missed components denoted by number(DM T C) is the number of elements remaining after noise removal operation. Figure 5 (c) shows the results of this detection and localization step.

EVALUATION OF OUR ALGORITHM

Quantitative evaluation

The proposed method has been evaluated on a set of 256 images taken randomly from Gallica's dataset. This set of selected images cover five centuries of printing to maximize the variety of fonts and physical structures. The images are either binary or gray levels images. Their resolution is 300dpi. The ground truth of our dataset was produced manually during a verification stage of the automatic OCR outputs provided in ALTO files, using a BnF tool that allows to correct the segmentation and the word transcription. The ground truth that we have is also in ALTO format.

The visual results (cf. figure 3) show that our system is able to identify correctly the missed text components as well as missed graphic elements ¶ . In some cases, some graphical components can be identified as textual elements when they have similar characteristics, such as in the case of separators between the textual elements (cf. figure 3(III)). But according to the visual results, our approach tends to make few over detection mistakes.

To evaluate quantitatively our approach, we first have to compute the True Missed Text Components (T M T C). This is done thanks to a matching process at image level between the segmentation performed by the OCR (in the ALTO output file) and the ground truth (also in ALTO). The matching is done according to this rule for each text component in the ground truth: if more than 80% of the surface of a text component in the ground truth (GT text ) is covered by text components in the OCR output, we consider that the text component is well detected by the OCR. Otherwise, the difference of the surface between each (GT text ) and the OCR text components output becomes a T M T C which is included in the image of the true missed text components (cf. figure 5 (a)). The number of textual components in this images is called: number(T M T C).

The detection rate of our approach is obtained using a matching process between the true missed text components previously obtained and the image of envelopes of the missed text components detected by our approach (T M T C ∩ DM T C). This process is rather different than classical segmentation algorithms since our aim is not the same: if more than 90% of a T M T C is covered by a DM T C, we consider that the corresponding T M T C is well detected. This way, our approach is able to detect 1886 missed text components (word or part of words) over the 2242 missed text components in the ground truth, which corresponds to a recall rate of 84.15% (cf. equation 9). We have also evaluated the false alarms (wrong detections) generated by our approach. In this ¶ One should remember that the aim of our approach is to detect the missed text components in the OCR outputs and not to operate an accurate segmentation.

This threshold is rather strict but avoids to consider partial detection of missed elements as true positives. case, in compliance with the visual results of the proposed approach, we can estimate that our approach has a good precision rate: 94.73% (cf. equation 10).

Recall = number (T M T C ∩ DM T C) number (T M T C) (9)
P recision = number (T M T C ∩ DM T C) number (DM T C) (10)

Potential use of the approach

Considering the good results of the approach, the system will be integrated into the OCR quality workflow at the BnF as an assistant tool for the OCR controlers. At present, the system is able to provide a visual feedback (cf. figure 4) to the controlers as well as the number of missed textual components (DTMC). The visual result include the original page image, the OCR elements (black boxes) and the missed elements detected by our approach (The green areas refer to the missed graphic elements and the red areas refer to the missed textual elements). With this feedback, the user will be able to build a quality report and to justify a potential rejection of the OCR outputs. Furthermore, the integration of our approach into the OCR quality workflow will give us the possibility to estimate the performance of the approach on a large dataset derived from the BnF workflow.

CONCLUSION AND PERSPECTIVES

This paper presents an original approach to control OCR output and provide quality assessment of digitization process by detecting missed text components in OCR outputs without using ground truth. Usually, OCR systems provide outputs with a good confidence rate considering the detected elements (text or graphics). Consequently, (a) (b) (c) these output files can be used to learn the specific properties of the components already detected (text and graphics). Moreover, they can be used to reduce the search space to detect missed text components, most of the time located outside the previously detected areas. This detection methodology based on page properties make our approach more robust to the diversity of styles and fonts that can be encountered in a wide range of documents, which has been verified considering the BnF collections.

The proposed approach uses 12 descriptors based on the Radon Transform, Local Binary Patterns and textual frequency descriptors to characterize pixels inside an image. Then, a SVM is used to classify the background pixels of the current image (those that were not previously classified by the OCR as text or graphic). Based on the classification results missed text components are finally localized.

Experimental results were performed on a heterogeneous set of images including old document images and new document images. They have shown the robustness of the proposed approach which has a good recall rate: 84.15% of missed text components are detected. Moreover, it does not cause many false alarms since it has a good precision rate of 94.73%. Nevertheless, our method has also some drawbacks. The main one is that we assume the areas already detected by the OCR are without errors. But, sometimes, errors can occur and then the classifier can learn wrong characteristics. To limit these defects we think to work not only at page level but also at book level or at least considering groups of images. Another thing we would like to investigate is to validate our experimental results by an empirical study to evaluate how our approach can be useful for people in charge of the control at the BnF.

Figure 2 .

 2 Figure 2. Case of a missing character in well segmented word

Figure 3 .

 3 Figure 3. Results of applying our approach of detecting missed components on the example of figure 1: red pixels indicate the missed text elements, green pixels indicate the missed graphical elements and blue pixels indicate the empty regions

Figure 4 .

 4 Figure 4. Detected missed components (Defects images)

Figure 5 .

 5 Figure 5. The needed information to conduct the evaluation of our approach:(a) The ground truth of the missed elements, (b) the images of distance transform of detected components. (c) The images of detected components.

‡ Graphical regions misclassified as textual regions and the reverse cases are assumed far less frequent, and do not fall within the scope of this study


Thomas Simon Total

Thomas Simon

TOTAL POSITIVITY OF A CAUCHY KERNEL

Keywords: 2000 Mathematics Subject Classification. 15A15, 15B48, 33C45, 60E07 . Alternating sign matrix -Cauchy kernel -Chebyshev polynomial -Izergin-Korepin determinant -Positive stable semi-group -Total positivity

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Let I be some real interval and K some real kernel defined on I × I. The kernel K is called totally positive of order n (TP n ) if det [K(x i , y j )] 1≤i,j≤m ≥ 0 for every m ∈ {1, . . . , n}, x 1 < . . . < x m and y 1 < . . . < y m . If these inequalities hold for all n one says that K is TP ∞ . The kernel K is called sign-regular of order n (SR n ) if there exists {ε m } 1≤m≤n ∈ {-1, 1} such that ε m det [K(x i , y j )] 1≤i,j≤m ≥ 0 for every m ∈ {1, . . . , n}, x 1 < . . . < x m and y 1 < . . . < y m . If these inequalities hold for all n one says that K is SR ∞ . The above four properties are called strict, with corresponding notations STP and SSR, when all involved inequalities are strict. We refer to [START_REF] Karlin | Total positivity[END_REF] for the classic account on this field and its various connections with analysis, especially Descartes' rule of signs. We also mention the recent monograph [START_REF] Pinkus | Totally positive matrices[END_REF] for a more linear algebraic point of view and updated references.

A function f : R → R + is called Pólya frequency of order n ≤ ∞ (PF n ) if the kernel K(x, y) = f (x-y) is TP n on R × R. When this kernel is STP n , we will use the notation f ∈ SPF n . Probability densities belonging to the class PF ∞ have been characterized by Schoenberg -see e.g. Theorem 7.3.2 (i) p. 345 in [START_REF] Karlin | Total positivity[END_REF] -through the meromorphic extension of their Laplace transform, whose reciprocal is an entire function having the following Hadamard factorization:

(1) 1 E[e sX ] = e -γs 2 +δs ∞ n=0

(1 + a n s)e -ans with X the associated random variable, γ ≥ 0, δ ∈ R, and a 2 n < ∞. The classical example is the Gaussian density. PF 2 functions are easily characterized by the log-concavity on their support -see Theorems 4.1.8 and 4.1.9 in [START_REF] Karlin | Total positivity[END_REF]. But except for the cases n = 2 or n = ∞ there is no handy criterion for testing the PF n character of a given function resp. the TP n character of a given kernel, and such questions might be difficult. In this paper we consider the kernel K α (x, y) = 1 x 2 + 2 cos(πα)xy + y 2 over (0, +∞) × (0, +∞), with α ∈ [0, 1). Because of its proximity with the standard Cauchy density we may call K α a Cauchy kernel, eventhough the denomination Poisson kernel would also be justified. Occurrences of the kernel K α in the literature are many and varied. We show the following Theorem. (a) One has

K α ∈ STP ∞ ⇔ K α ∈ SR ∞ ⇔ α ∈ {1/2, 1/3, . . . , 1/n, . . . , 0}.
(b) For every n ≥ 2, one has

K α ∈ TP n ⇔ K α ∈ SR n ⇒ α ∈ {1/2, 1/3, . . . , 1/n} or α < 1/n.
(c) For every n ≥ 2, one has α ≤ 1/n ∧ 1/(n 2 -n -6) + ⇒ K α ∈ STP n .

The well-known fact that K 0 and K 1/2 are STP ∞ is a direct consequence of explicit classical formulae for the involved determinants, due respectively to Cauchy and Borchardt -see e.g. (2.7) and (3.9) in [START_REF] Krattenthaler | Advanced determinant calculus[END_REF] -and which will be recalled thereafter. The characterization obtained in Part (a) follows without difficulty from Schoenberg's theorem and Gauss' multiplication formula, and will be given in Section 2. The more involved proofs of Part (b) and Part (c) rely both on an analysis of the derivative determinant

∆ n α = det n ∂ i+j-2 K α ∂x i-1 ∂y j-1
where, here and throughout, we set det n for the determinant of a matrix whose rows and columns are indexed by (i, j) ∈ {1, . . . , n} 2 . To obtain Part (b) we establish a closed expression for ∆ n α (1, 0+) in terms of Chebyshev polynomials of the second kind, an expression which is negative whenever α ∈ {1/2, 1/3, . . . , 1/n} or α > 1/n. The computations are performed in Section 3, in four manners involving respectively Wronskians, Schur functions, rectangular matrices and alternating sign matrices. The observation that these four very different approaches all lead to the same formula was interesting to the author. On the other hand only the last approach, which is based on Propp's formula for the Izergin-Korepin determinant, seems successful to get the more difficult Part (c). This latter result is partial because our upper condition on α is probably not optimal. We raise the Conjecture. For every n ≥ 2, one has

(2) α < 1/n ⇒ K α ∈ STP n .
This would show that the inclusion in Part (b) is actually an equivalence, which is true for n = 2, 3 by Part (c) of the theorem because then n ≥ (n 2 -n -6) + . In Section 5 we show [START_REF] Bhatia | Positive definite matrices[END_REF] for n = 4, 5, and we also give some heuristics reasons supporting the validity of this inclusion for all n. The latter seems however to require wild computations. In Section 5, we state three other conjectures of combinatorial nature whose fulfilment would entail [START_REF] Bhatia | Positive definite matrices[END_REF]. The most natural one is a criterion for the positivity of the generating function

f n,k (z) = A ∈ An µ(A) = k z ν(A) zν(A Q )
evaluated at a certain complex number z, where A n is the set of alternating sign matrices of size n, A Q is the anticlockwise quarter-turn rotation of A, µ(A) is the number of negative entries and ν(A)

the inversion number (see the precise notations below). Whereas we can show this criterion for k = 0 or k = µ max (see below Propositions 3 and 4), unfortunately we cannot do so for all k since there is no sufficiently explicit general formula for f n,k . Let us stress that the single evaluation of

f n,k (1) = ♯{A ∈ A n , µ(A) = k} is a difficult
open problem, solved only for certain values of k -see [START_REF] Gac | Quelques problèmes d'énumération autour des matrices à signe alternant[END_REF] and the references therein.

The present paper was initially motivated by a question of S. Karlin -see Section 6 below for its precise statement -on the total positivity in space-time of the positive stable semi-group (t, x) → p α (t, x) on (0, +∞) × (0, +∞), which we recall to be defined by the Laplace transform

(3) ∞ 0 p α (t, x)e -λx dx = e -tλ α , λ ≥ 0.
As a simple consequence of Part (b), in Section 6 we show the Corollary. For every n ≥ 2, one has

p α ∈ SR n ⇒ α ∈ {1/2, 1/3, . . . , 1/n} or α < 1/n.
The inclusion α ≤ 1/n ⇒ p α ∈ TP n was proved in [START_REF] Simon | Multiplicative strong unimodality for positive stable laws[END_REF] for n = 2 and we believe that it is true in general. This would give a complete answer to Karlin's question and also entail the above conjecture -see Section 6 for an explanation. This will be the matter of further research.

Proof of part (a)

This easy part of the theorem is a consequence of the following Proposition 1 and Corollary 1.

Proposition 1. One has

K α ∈ STP ∞ ⇔ K α ∈ TP ∞ ⇔ α ∈ {1/2, 1/3, . . . , 1/n, . . . , 0}.
Proof. We begin with the second equivalence. Consider the generalized logistic distribution with density g α (x) = sin(πα) 2πα(cosh(x) + cos(πα)) over R. Simple transformations -see Theorem 1.2.1. in [START_REF] Karlin | Total positivity[END_REF] -entail that the TP ∞ character of K α amounts to the fact that g α ∈ PF ∞ . For every s ∈ (-1, 1), compute

R e sx g α (x) dx = sin(πα) πα ∞ 0 u s u 2 + 2u cos(πα) + 1 du = sin(παs) α sin(πs)
where the right-hand side follows from the residue theorem and is meant as a limit for α = 0. If

α / ∈ {1/
= Γ(1 -s)Γ(1 + s) Γ(1 -s/n)Γ(1 + s/n)
and applying Gauss' multiplication formula and Weierstrass formula for the Gamma function -see e.g. 1.2(11) p.4 and 1.1(3) p.1 in [START_REF] Erdélyi | Higher transcendental functions[END_REF] -shows that this function is of the type [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF], in other words that g α ∈ PF ∞ . If α = 0, the same conclusion holds true thanks to the Eulerian formula sin(πs)

πs = n≥1 1 - s 2 n 2 •
This finishes the proof of the second equivalence. To show the first one, it remains to prove that K α ∈ STP ∞ whenever α ∈ {1/2, 1/3, . . . , 1/n, . . . , 0}. Cauchy's double alternant formula (see e.g.

(2.7) in [START_REF] Krattenthaler | Advanced determinant calculus[END_REF] or Example 4.3 in [START_REF] Pinkus | Totally positive matrices[END_REF])

det n 1 x 2 i + y 2 j = 1≤i<j≤n (y 2 j -y 2 i )(x 2 j -x 2 i )
1≤i,j≤n (x 2 i + y 2 j ) entails immediately that K 1/2 ∈ STP ∞ . Analogously, Borchardt's formula (see e.g. (3.9) in [START_REF] Krattenthaler | Advanced determinant calculus[END_REF])

(5) det n 1 (x i + y j ) 2 = det n 1 x i + y j × perm n 1 x i + y j yields K 0 ∈ STP ∞ .
An alternative way to prove these two latter facts consists in writing 1 x 2 + y 2 = ∞ 0 e -x 2 u e -uy 2 du and 1 (x + y) 2 = ∞ 0 e -xu e -uy udu.

Indeed, the composition formula -see Lemma 3.1.1 in [START_REF] Karlin | Total positivity[END_REF] -entails that K 0 and K 1/2 are STP ∞ because the kernel e -ux is SRR ∞ on (0, +∞) × (0, +∞) -see [START_REF] Karlin | Total positivity[END_REF] p. 18 for the latter fact and p. 12 for an explanation of the notation SRR ∞ .

The remaining cases α ∈ {0, 1/2} are slightly more involved. First, it follows from (4) and a fractional moment identification that f 1/n is the density of the independent sum [START_REF] Gessel | Binomial determinants, paths, and hook-length formulae[END_REF] log(Γ

1/n ) + • • • + log(Γ (n-1)/n ) -log(Γ 1/n ) -• • • -log(Γ (n-1)/n )
where Γ t is for every t > 0 the random variable with density

x t-1 e -x Γ(t) 1 {x>0} .
Second, it is straightforward that the density of log(Γ t ) is SPF ∞ . This property conveys to f 1/n thanks to the above factorization and the composition formula. (b) It is easy to see that x → K α ( √ x, √ y) is a completely monotone function for every y > 0, and that there exists a certain positive finite kernel L α (x, y) on (0, +∞) × (0, +∞) such that

K α (x, y) = ∞ 0 e -x 2 u L α (u, y)du.
When α is the reciprocal of an integer, by [START_REF] Gessel | Binomial determinants, paths, and hook-length formulae[END_REF] it is possible to express L α as a convolution of weighted Laplace transformation kernels and show that it is RR ∞ . The kernel L α is less explicit in the other cases but we feel that its sign-regularity index matches the total positivity index of K α , in other terms that

K α ∈ TP n ⇔ L α ∈ RR n for all n.
Proposition 2. The function g α is positive-definite for every α ∈ [0, 1).

Proof. The beginning of the proof of Proposition 1 entails by analytic continuation that R e isx g α (x) dx = sinh(παs) α sinh(πs)

for every s ∈ R. Since α ∈ [0, 1), we can apply the Fourier inversion formula to obtain

g α (x) = R e isx sinh(παs) 2πα sinh(πs) ds, x ∈ R.
The conclusion follows from Bochner's theorem.

Remark 2. The above proposition is very well-known, but we gave a proof for the reader's comfort.

It is also true that g r α is positive-definite for every α ∈ [0, 1) and r > 0 -see e.g. Exercise 5.6.22 (ii) in [START_REF] Bhatia | Positive definite matrices[END_REF].

Corollary 1. For every α ∈ [0, 1) and n ≥ 2, one has

K α ∈ TP n ⇔ K α ∈ SR n .
Proof. Proposition 2 entails that for every α ∈ [0, 1) the function

x → 1 + cos(πα) cosh(x) + cos(πα)
is the characteristic function of the random variable X α with density t →

(1 + cos(πα)) sinh(παt) sin(πα) sinh(πt) •

For every n ≥ 2, s 1 , . . . , s n ∈ R and 0 < x 1 < . . . < x n , this yields 1≤i,j≤n

s i s j K α (x i , x j ) = 1 2(1 + cos(πα)) E   n k=1 t k e iy k Xα 2   > 0,
with the notation t i = s i /x i and y i = log(x i ). Hence, the quadratic form [K α (x i , x j )] 1≤i,j≤n is positive definite for every n ≥ 2 and 0 < x 1 < . . . < x n . In particular, one has

det [K α (x i , x j )] 1≤i,j≤n > 0,
for every n ≥ 2 and 0 < x 1 < . . . < x n . This shows the required equivalence.

Proof of part (b)

Our argument for this part is the same as the one we have used in [START_REF] Simon | Produit Beta-Gamma et régularité du signe[END_REF], in the framework of confluent hypergeometric functions. Let X = {x 1 < . . . < x n } and Y = {y 1 < . . . < y n } be two sets of positive variables and

V X = 1≤i<j≤n (x j -x i ) and V Y = 1≤i<j≤n (y j -y i ) be the usual Vandermonde determinants. Consider D n α (X, Y ) = det n [K α (x i , y j )
] and the aforementioned derivative determinant

∆ n α (x, y) = det n ∂ i+j-2 K α ∂x i-1 ∂y j-1 (x, y) , x, y > 0.
Using repeatedly the formula

p k=0 (-1) p-k C k p f (z + kε) ∼ f (p) (z)ε p , ε → 0+
which is valid for any smooth real function f , elementary operations on rows and columns show that

(7) ∆ n α (x, y) = sf(n -1) 2 lim ε→0+ D n α (X ε , Y ε ) V Xε V Yε
where we have set

x ε i = x + (i -1)ε, y ε i = y + (i -1)ε for i = 1 . . . n, X ε = (x ε 1 , . . . , x ε n ), Y ε = (y ε 1 , . . . , y ε n ), and sf(k) = k i=0 i!
for the superfactorial number. By Proposition 2, this entails that ∆ n α (x, x) ≥ 0 for any x > 0 and below it will be established that the inequality is actually everywhere strict -see Remark 5.

On the other hand, if ∆ k α (1, 0+) < 0 for some k ∈ {2, . . . , n}, then [START_REF] Karlin | Total positivity[END_REF] entails that K α is not TP n and hence not SR n by Corollary 1. We will prove that this is the case as soon as α ≥ 1/n and α / ∈ {1/2, 1/3, . . . , 1/n}. More precisely, setting

U α k = sin kπα sin πα and V α n = n k=1 U α k for every k, n ≥ 1, we will show that (8) ∆ n α (1, 0+) = sf(n -1) 2 V α n .
We give four different proofs of [START_REF] Krattenthaler | Advanced determinant calculus[END_REF], each corresponding to a specific approach to evaluate the derivative determinant ∆ n α (x, y) in closed form. The first two proofs involve classical tools, respectively Wronskians and Schur functions. The last two proofs rely on more elaborate and recent results on the Izergin-Korepin determinant, using combinatorial formulae due respectively to Lascoux and Propp. The latter formula, involving alternating sign matrices, will be also used to obtain Part (c).

3.1. Wronskians. We fix α ∈ [0, 1), set c α = cos(πα) and s α = sin(πα). We use the notations

Q α (x, y) = y + x cos(πα), Q α (x, y) = Q α (y, x), R α (x, y) = Q α (x, y) Q α (x, y),
and P α (x, y) = x 2 + 2xy cos(πα) + y 2 . In the following, we will often skip the dependence in (x, y)

for concision. Observe that (9) R α -c α P α = s 2 α xy and Q 2 α -P α = -s 2 α y 2 .
Introduce the kernels

T α,2p = (2p)! p k=0 (-1) p-k C p-k p+k (4Q 2 α ) k K p+k+1 α and T α,2p+1 = 2(2p + 1)! Q α p k=0 (-1) p+1-k C p-k p+1+k (4Q 2 α ) k K p+k+2 α
for any integer p.

Lemma 1. For every n ≥ 0, one has

∂ n K α ∂y n = T α,n .
Proof. The formula is obviously true for n = 0, 1. By induction, it suffices to show that

∂T α,2p ∂y = T α,2p+1 and ∂T α,2p+1 ∂y = T α,2p+2
for any integer p. The latter are elementary computations, whose details are left to the reader.

Lemma 1 allows to express ∆ n α as the Wronskian of (T α,0 , . . . , T α,n-1 ) built on x-derivatives, in other words one has [START_REF] Gac | Quelques problèmes d'énumération autour des matrices à signe alternant[END_REF] ∆

n α = det n ∂ i-1 T α,j-1 ∂x i-1 .
Observe that

T α,2p (1, 0+) = (2p)! p k=0 (-1) k C k 2p-k (2 cos(πα)) 2p-2k = (2p)! sin((2p + 1)πα) sin(πα)
where the second equality comes after identifying two standard definitions of the Chebyshev polynomial of the second kind U 2p . Making the same identification for T α,2p+1 (1, 0+) and putting the two together entail [START_REF] Pinkus | Totally positive matrices[END_REF] T α,r (1, 0+) = (-1) r r! sin((r + 1)πα) sin(πα) for every r ≥ 0. We next compute the successive x-derivatives of T α,r at (1, 0+). The outline of the proof is analogous to Lemma 1, but it is more involved and so we give the details.

Lemma 2. For every j ≥ 1 and r ≥ 0, one has

(12) ∂ j T α,r ∂x j (1, 0+) = (-1) j (r + j + 1)! (r + 1)! T α,r (1, 0+).
Proof. We first consider the case r = 2p. Setting

a j,r = (r + j + 1)! (r + 1)! , E α j,p = (-1) j (2p)! ∂ j T α,2p ∂x j , F α k,p = (-1) p-k C p-k p+k 4 k Q 2k-1 α , G α k,p = Q α F α k,p ,
we will show by induction that for every j ≥ 1, one has

(13) E α j,p = a j,r Q j α p k=0 G α k,p K p+k+j+1 α + yH α j,p
for some smooth function H α j,p on (0, ∞) × [0, ∞). Specifying to (x, y) = (1, 0+) clearly entails [START_REF] Simon | Multiplicative strong unimodality for positive stable laws[END_REF]. We begin with the case j = 1, in computing

E α 1,p = 2 p k=0 (p + k + 1)F α k,p R α K p+k+2 α -c α p k=1 kF α k,p P α K p+k+2 α = (2p + 2) Q α p k=0 G α k,p K p+k+2 α + 2xys 2 α p k=1 F α k,p K p+k+2 α = (2p + 2) Q α p k=0 G α k,p K p+k+2 α + yH α 1,p
for some smooth function

H α 1,p on (0, ∞) × [0, ∞)
, where in the second equality we used the first identity in [START_REF] Lascoux | Square ice enumeration[END_REF]. Suppose now that (13) holds for some j ≥ 1. Differentiating, we obtain

E α j+1,p = a j,r Q j-1 α p k=0 (2(p + k + j + 1)G α k,p Q 2 α -2kc α P α F α k,p Q α -jG α k,p )K p+k+j+2 α -y ∂H α j,p ∂x = (2p + j + 2)a j,r Q j+1 α p k=0 G α k,p K p+k+j+2 α + 2(R α -c α P α ) Q j α p k=0 kF α k,p K p+k+j+2 α + ( Q 2 α -P α ) Q j-1 α p k=0 G α k,p K p+k+j+2 α -y ∂H α j,p ∂x = a j+1,r Q j+1 α p k=0 G α k,p K p+k+j+2 α + yH α j+1,p
for some smooth function H α j+1,p on (0, ∞) × [0, ∞), where in the second equality we used the two identities in [START_REF] Lascoux | Square ice enumeration[END_REF]. This completes the proof of (13) for r = 2p. An analogous induction shows that (-1) j (2p + 1)!

∂ j T α,2p+1 ∂x j = 2 a j,2p+1 Q α Q j α p k=0 (-1) p+1-k C p-k p+1+k (4Q 2 α ) k K p+k+j+2 α + yI α j,p
for every j ≥ 1, p ≥ 0, where I α j,p is a smooth function on (0, ∞) × [0, ∞). This yields (12) for r = 2p + 1, and finishes the proof.

We finally deduce from [START_REF] Gac | Quelques problèmes d'énumération autour des matrices à signe alternant[END_REF], [START_REF] Pinkus | Totally positive matrices[END_REF], Lemma 2, and Leibniz's formula for determinants, that

∆ n α (1, 0+) = sf(n -1) V α n det n (i + j -1)! j! = sf(n -1) 2 V α n det n C j i+j-1 = sf(n -1) 2 V α n ,
where the last equality follows from the standard evaluation of a binomial determinant which is left to the reader. This completes the proof of (8).

Remark 3. It does not seem that [START_REF] Gac | Quelques problèmes d'énumération autour des matrices à signe alternant[END_REF] and standard Wronskian transformations can provide any information which is enough explicit to characterize the everywhere positivity of (x, y) → ∆ n α (x, y). The latter is crucial for (2) -see Section 4 below.

3.2. Schur functions. This second approach relies on the following well-known expression for the generating function of Chebyshev polynomials of the second kind:

k≥0 U α k+1 z k = K α (1, -z), |z| < 1.
This entails

D n α (X, Y ) = 1 n j=1 x 2 j det n   k≥0 U α k+1 (y i z j ) k  
with the above notation for X and Y, having written z j = -x -1 j and assuming, here and throughout, |y i z j | < 1 for all i, j = 1, . . . , n. The determinant on the right-hand side, denoted by F n α (X, Y ), can be further evaluated by multilinear expansion:

F n α (X, Y ) = ∞ k 1 ...kn=0 det n z k i j n l=1 U α k l +1 y k l l .
Notice that in the multiple sum, all indices k i can be chosen distinct since otherwise the summand is zero. Setting S n for the symmetric group of size n and

U α K = n l=1 U α k l +1
for any n-tuple K = (k 1 , . . . , k n ), one obtains

F n α (X, Y ) = k 1 >•••>kn≥0 U α K σ∈Sn det n z k σ(i) j n l=1 y k σ(l) l = k 1 >•••>kn≥0 U α K det n z k i j det n y k i j ,
where in the second equality we used twice Leibniz's formula. The above can now be rewritten in terms of Schur functions and Vandermonde determinants. Setting k i = n -i + λ i , one has

F n α (X, Y ) = λ 1 ≥•••≥λn≥0 U α K det n z n-i+λ i j det n y n-i+λ i j = V Y V Z λ U α K s λ (Y )s λ (Z)
where in the second equality the sum is meant on all partitions of size n and we use the standard notation for Schur functions, displayed e.g. in Chapter 4 p.124 of [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]. Putting everything together and differentiating with (7) yield after some simplifications

∆ n α (x, y) = sf(n -1) 2 x n(n+1)/2 λ n i=1 U α n-i+λ i +1 s λ (1, . . . , 1) 2 (-yx -1 ) |λ| for all 0 < y < x, with the notation |λ| = λ 1 + • • • + λ n .
By the change of variable µ i = λ n+1-i , the classical formula s λ (1, . . . , 1) = 1≤i<j≤n (λ i -λ j + j -i) sf(n -1) which can be recovered from the Jacobi-Trudi identity -see e.g. Proposition 4.2 in [3] -and a standard binomial determinant evaluation, entails finally

(14) ∆ n α (x, y) = 1 x n(n+1)/2 ∞ k=0 (-yx -1 ) k µ1 ≤ • • • ≤ µn µ1 + • • • + µn = k n i=1 U α i+µ i 1≤i<j≤n (µ j -µ i + j -i) 2 .
We can now compute explicitly ∆ n α (1, 0+) because the summation is made on the single partition (0, . . . , 0) : we get

∆ n α (1, 0+) = n i=1 U α i × 1≤i<j≤n (j -i) 2 = sf(n -1) 2 V α n ,
as required by [START_REF] Krattenthaler | Advanced determinant calculus[END_REF].

Remark 4. Because of the alternate signs, the above expression (14) does not seem very helpful either to study the everywhere positivity of ∆ n α (x, y).

3.3.

Rectangular matrices. This third approach hinges upon a certain closed expression for the determinant

D q (X, Y ) = det n 1 (x i + y j )(qx i + y j )
.

The latter is called the Izergin-Korepin determinant in the literature, and appears in the context of the six-vertex model -see Chapter 7 in [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]. We use its evaluation in terms of rectangular matrices separating the variables, which is due to Lascoux -see Theorem q in [START_REF] Lascoux | Square ice enumeration[END_REF]. It is given by

D q (X, Y ) = V X V Y P q (X, Y ) det n H X × E q Y
where P q (X, Y ) = 1≤i,j≤n (x i + y j )(qx i + y j ), and

H X = [h k-i (X)] 1≤i≤n,1≤k≤2n-1 , E q Y = q k-j+1 -q j-1 q -1 e n-k+j-1 (Y ) 1≤k≤2n-1,1≤j≤n
are two rectangular matrices involving the complete resp. elementary symmetric functions, which we recall to be defined through the generating functions

(15) k≥0 h k (X)t k = n r=1 (1 -x r t) -1 and k≥0 e k (Y )t k = n r=1
(1 + y r t).

Setting q = e 2iπα , X q x = (xq -1/2 , . . . , xq -1/2 ), and Y y = (y, . . . , y), Lascoux's formula and ( 7) yield

∆ n α (x, y) = sf(n -1) 2 q -n(n-1)/4 K α (x, y) n 2 det n H X q x × E q Yy .
By (15), we have h r (X q x ) = C r n+r-1 x r q -r/2 , whence

H X q x = C k-i n+k-1-i q (i-k)/2 x k-i 1≤i≤n,1≤k≤2n-1
.

On the other hand, (15) entails e r (Y y ) = C r n y r , so that after some simplifications

E Yy = C n-k+j-1 n q (k-1)/2 U α k+2-2j y n-k+j-1 1≤k≤2n-1,1≤j≤n
, with our above notation for U α r . The i-th row of the product H X q

x × E Yy having a factor q (i-1)/2 , we finally obtain

(16) ∆ n α (x, y) = sf(n -1) 2 K α (x, y) n 2 det n [A n (x) × B α n (y)]
with the notations

A n (x) = C k-i n+k-1-i x k-i 1≤i≤n,1≤k≤2n-1 and B α n (y) = C n-k+j-1 n U α k+2-2j y n-k+j-1 1≤k≤2n-1,1≤j≤n
.

Setting L α n (x, y) = det n [A n (x) × B α n (y)] , the Cauchy-Binet formula entails (17) L α n (x, y) = 1≤σ 1 <...<σn≤2n-1 A σ (x)B α σ (y)
where A σ (x) is the n × n minor obtained from the columns σ 1 , . . . , σ n in A n (x) and B α σ (y) is the n × n minor obtained from the rows σ 1 , . . . , σ n in B α n (y). By Leibniz's formula, one has

A σ (x) = A σ (1)x nσ and B α σ (y) = B α σ (1)y n(n-1)-nσ
with the notation

n σ = n i=1 (σ i -i).
This shows that L α n (x, y) is a homogeneous polynomial of degree n(n -1) with coefficient (18

) nσ=k A σ (1)B α σ (1)
for the term x k y n(n-1)-k . Besides, by ( 16) and symmetry, we have L α n (x, y) = L α n (y, x) so that these coefficients are palindromic. We compute

L α n (1, 0+) = L α n (0+, 1) = A Ĩd (1)B α Ĩd (1)
with the notation Ĩd = (n, . . . , 2n -1). One finds immediately B α

Ĩd (1) = U α 1 × • • • × U α n and

some elementary linear transformations yield A α

Ĩd (1) = 1. Since K α (0+, 1) = 1, we finally deduce (8).

3.4. Alternating sign matrices. In this last approach we use Izergin-Korepin's original formula for the determinant D q (X, Y ) and its expression in terms of alternating sign matrices, due to Propp, which is given in Exercise 7.2.13 p. 244 in [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]. It reads

(19) D q (X, Y ) = V X V Y P q (X, Y ) A∈An (-1) µ(A) (1 -q) 2µ(A) q C 2 n -I(A) n i=1 x µ i (A) i y µ i (A) i 1 ≤ i, j ≤ n aij = 0 (α ij x i + y j )
where A n stands for the set of n × n alternating sign matrices (ASM) viz. those matrices made out of 0s, 1s and -1s for which the sum of the entries in each row and each column is 1, and the non-zero entries in each row and column alternate in sign. We refer to [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF] for a comprehensive account on this topic, and also to Section 3 in the recent paper [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF] for updated results. In (19), the following notations are used: µ i (A) resp. µ i (A) is the number of -1s in the i-th row resp. i-th column of A, µ(A) the total number of -1s in A, I(A) the generalized inversion number of A viz.

I(A) = i<k,l<j a ij a kl , and
α ij = q if k≤i a kj = l≤j a il , 1 otherwise.
Notice that ASM matrices without -1 are permutation matrices. In particular, one recovers Borchardt's formula ( 5) from (19) in setting q = 1 -see Exercise 7.2.14 in [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]. In the following we will set J(A) = ♯{(i, j), k≤i a kj = l≤j a il }, which is the number of southwest or northeast molecules in the terminology of the six-vertex model, and satisfies the formula (20)

J(A) = 2I(A) -2µ(A)
(see Exercise 7.1.8 in [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]). Setting q = e 2iπα , we see that [START_REF] Karlin | Total positivity[END_REF] entails after some simplifications hinging upon (20) the following expression for the derivative determinant ∆ n α (x, y):

(21) sf(n -1) 2 K α (x, y) n 2 A∈An (P α (x, y)) µ(A) (Q α (x, y)) J(A) ( Qα (x, y)) n(n-1)-2I(A)
with the notations

P α (x, y) = 4 sin 2 (πα)xy and Q α (x, y) = e iπα/2
x + e -iπα/2 y.

This yields

∆ n α (1, 0+) = sf(n -1) 2 e -iπn(n-1)α/2 A∈Sn e 2iπαI(A) ,
where S n stands for the set of permutation matrices of size n. Using the generating function of I(A)

for permutation matrices given e.g. in Corollary 3.5 of [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF], further trigonometric simplifications entail

∆ n α (1, 0+) = sf(n -1) 2 V α n ,
as required by [START_REF] Krattenthaler | Advanced determinant calculus[END_REF].

Remark 5. The formula (21) also shows that

∆ n α (x, x) = sf(n -1) 2 (2x cos(πα/2)) n(n+1)
A∈An (4 sin 2 (πα/2)) µ(A) > 0 for all x > 0 and α ∈ [0, 1).

Proof of part (c)

This last part of the theorem relies on Karlin's ETP criterion on the derivative determinant (see Theorem 2.2.6 in [START_REF] Karlin | Total positivity[END_REF]) which states that if ∆ k α (x, y) > 0 for every k ∈ {2, . . . , n} and x, y > 0, then K α is STP n . By an induction, we hence need to show that

(22) α ≤ 1/n ∧ 1/(n 2 -n -6) + ⇒ ∆ n α (x, y) > 0 for all x, y > 0.
This will be obtained from (21) and some considerations on ASM matrices, all to be found in [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF] and Section 2.1 of [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]. Again, for concision we will skip the dependence of the involved kernels in (x, y). For any A ∈ A n , introduce the statistics

ν(A) = 1 ≤ i < i ′ ≤ n 1 ≤ j < j ′ ≤ n A ij A i ′ j ′ = J(A) 2
(see pp. 5-6 in [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF] for an explanation of the second equality). Set µ n = max{µ(A), A ∈ A n } and notice that µ n = (n -1) 2 /4 for n odd and that µ n = n(n -2)/4 for n even (see again p. 6 in [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]).

From (21), it is clear that if (23) F α n,k = A ∈ An µ(A) = k Q 2ν(A) α Qn(n-1)-2ν(A)-2k α > 0
for all k = 0 . . . µ n , then ∆ n α is everywhere positive. Notice first that F α n,k is real since it writes 1 2

A ∈ An µ(A) = k (Q 2ν(A) α Qn(n-1)-2ν(A)-2k α + Q2ν(A) α Q n(n-1)-2ν(A)-2k α ).
Indeed, setting A Q for the anticlockwise quarter-turn rotation of A, one has

A Q ∈ A n with µ(A Q ) = µ(A) and 2ν(A Q ) = n(n -1) -2ν(A) -2µ(A) (see p. 7 in [1]).
Suppose now that k = 0. Then necessarily n ≥ 3, ν(A) ≥ 1 and ν(A Q ) ≥ 1 (see p. 7 in [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]) and one gets the factorization

F α n,k = |Q α | 4 A ∈ An µ(A) = k ℜ (Q 2(ν(A)-1) α Q2(ν(A Q )-1) α ) = |Q α | 4 ν n,k i=1 a i n,k ℜ (Q 2(i-1) α Qn(n-1)-2k-2(i+1) α )
where a i n,k are non-negative integer coefficients and ν n,k = max{ν(A), A ∈ A n , µ(A) = k}. Notice in passing that no closed expression for a i n,k or even ν n,k seem available in the literature. For n = 3, necessarily k = 1 = ν 3,1 and there is only one summand, so that

F α 3,1 = |Q α | 4 > 0. For n ≥ 4, it is sufficient to check that ℜ (Q 2(i-1) α Qn(n-1)-2k-2(i+1) α ) > 0 for all i = 1 . . . ν n,k . Noticing that n(n -1) -2k -2(i + 1) ≤ n 2 -n -6 and recalling that Q α (x, y) = e iπα/2
x + e -iπα/2 y, this inequality becomes clearly true when α ≤ 1/(n 2 -n -6) after expanding the trigonometric polynomials, because all involved cosines are evaluated inside [0, π/2].

It remains to consider the case k = 0. Reasoning exactly as above, for every n ≥ 2 one obtains F α n,0 > 0 whenever α ≤ 1/n(n -1). The following proposition yields the optimal condition α ≤ 1/n. Together with the above discussion, it concludes the proof of Part (c). Proposition 3. For every n ≥ 2, one has F α n,0 > 0 whenever α ≤ 1/n.

Proof. Using again the generating function of the inversion numbers of permutation matrices, we obtain the explicit formula

F α n,0 = n i=1 Q2i α -Q 2i α Q2 α -Q 2 α .
By an induction argument it is hence sufficient to prove that

α ≤ 1/n ⇒ Q2n α -Q 2n α Q2 α -Q 2 α > 0
for every n ≥ 2. Setting a = e 2iπα and recalling the definition of Q α , we write

Q2n α -Q 2n α Q2 α -Q 2 α = p+q=n-1 2p j=0 2q k=0 (a 1/4 x) j (a -1/4 y) 2p-j (a -1/4 x) k (a 1/4 y) 2p-k = 2n-2 i=0 c i x i y 2n-2-i
for some palindromic sequence {c i , i = 0, . . . , 2n -2} which is given by

c i = p+q = n -1 j +k = i j ≤ 2p k ≤ 2q a (j-k+q-p)/2 .
Summing appropriately and using some standard trigonometry, one can show that

c i = cos(π(n -1 -i)α) -cos(πnα) sin 2 (πα)
for every i = 0, . . . , 2n-2. We omit the details. This completes the proof since the latter expressions are all positive whenever α ≤ 1/n. Notice also that these formulae show that the sequence {c i } is unimodal.

Open questions

In this section we give some heuristic reasons supporting the validity of the inclusion (24) α < 1/n ⇒ ∆ n α (x, y) > 0 for all x, y > 0 which, by the ETP criterion and an immediate induction, is enough to show [START_REF] Bhatia | Positive definite matrices[END_REF]. First, we formulate a conjecture on the positivity of certain generating functions of ASM matrices which would entail (24), and we test it on the values n = 3, 4, 5. Second, we discuss more thoroughly Lascoux's factorization and settle two problems on the positivity on certain minors of rectangular matrices involving Chebyshev polynomials, whose solution would again entail (24).

5.1. Alternating sign matrices. By (21), the positivity of ∆ n α amounts to that of

A∈An P µ(A) α Q 2ν(A) α Q2ν(A Q ) α ,
with the notations of Section 4. The above sum can be expressed with the bivariate generating function

Z n (x, y) = A∈An x ν(A) y µ(A) ,
which is itself given as a certain functional determinant -see Formula (57) p. 21 in [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]. Unfortunately the latter determinant is in general very difficult to evaluate, except in certain particular cases. Its value at x = y = 1, counting the cardinal of A n , was the matter of a whole story which is told in the book [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]. As in the proof of Part (c) let us now write

A∈An P µ(A) α Q 2ν(A) α Q2ν(A Q ) α = µn k=0 P µ(A) α F α n,k ,
recalling the notations µ n = max{µ(A), A ∈ A n } and

F α n,k = A ∈ An µ(A) = k Q 2ν(A) α Q2ν(A Q ) α .
In view of Proposition 3, it is natural to raise the following question.

Conjecture 1. For every n ≥ 2 and every k = 0, . . . , µ n , one has F α n,k > 0 whenever α ≤ 1/n.

The kernel F α n,k can be expressed in terms of the generating function

Z n,k (x) = 1 k! ∂ k Z n ∂y k (x, 0
) but unfortunately no tractable closed formula is known for the latter. Even its value at x = 1, which counts the number of elements of A n with k negative entries, is known in closed form only in some cases -see Chapter 3 in [START_REF] Gac | Quelques problèmes d'énumération autour des matrices à signe alternant[END_REF] and the references therein.

By Proposition 3, Conjecture 1 is true for k = 0. In the preceding section, we also proved that F α 3,1 > 0 whenever α ≤ 1/3. Let us now show the validity of Conjecture 1 for n ≥ 4 and k = µ n . Recall that µ 2 = 0 and µ 3 = 1 so that there is no loss of generality in considering n ≥ 4. Proposition 4. For every n ≥ 4 one has F α n,µn > 0 whenever α ≤ 1/n.

Proof. First, suppose that n = 2p + 1 is odd. Then µ n = p 2 and this concerns one single matrix with inversion number ν = p(p + 1)/2 (see p. 6 in [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]). We deduce

Z n,µn (x) = x p(p+1)/2 and F α n,µn = |Q α | 2p(p+1) > 0.
Second, suppose that n = 2p is even. Then µ n = p(p -1) and this concerns two matrices with inversion numbers ν = p(p + 1)/2 resp. ν = p(p -1)/2 (see again p. 6 in [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]). We deduce

Z n,µn (x) = x p(p-1)/2 (1 + x p ) and F α n,µn = |Q α | 2p(p-1) (Q n α + Qn α ) > 0 if α ≤ 1/n.
Since there is no closed formula for F α n,k if k ∈ {0, µ n }, it seems quite difficult to establish Conjecture 1 for all n, k. Let us conclude this paragraph in checking its validity for n = 4 and n = 5. Since µ 4 = 2 and µ 5 = 4 we just have to consider the cases k = 1 resp. k = 1, 2, 3.

• Z 4,1 (x) = 2x(1 + x) 3 and F α 4,1 = 2|Q α | 4 (Q 2 α + Q2 α ) 3 > 0 if α ≤ 1/2. • Z 5,1 (x) = x(3 + 14x + 35x 2 + 48x 3 + 48x 4 + 35x 5 + 14x 6 + 3x 7 ), which rewrites 3x x 5 -1 x 3 -1 + 8x 2 x 4 -1 x 2 -1 + 10x 3 x 4 -1 x 3 -1 + 2x 4 x 2 -1
x -1 and yields

F α 5,1 = 3|Q α | 4 F α 5,0 F α 3,0 + 8|Q α | 8 F α 4,0 F α 2,0 + 10|Q α | 12 F α 4,0 F α 3,0 + 2|Q α | 16 F α 2,0 .
The latter is positive for α ≤ 1/5, by Proposition 3.

• Z 5,2 (x) = x(2 + 12x + 21x 2 + 24x 3 + 21x 4 + 12x 5 + 2x 6 ), which rewrites 2x(1 + x)(1 + x + x 2 )(1 + x + x 2 + x 3 ) + 6x 2 (1 + x 2 ) 2 + 11x 3 (1 + x 2 )
and yields

F α 5,2 = 2|Q α | 4 F α 4,0 + 6|Q α | 8 (Q 4 α + Q4 α ) 2 + 11|Q α | 12 (Q 4 α + Q4 α )
. The latter is positive for α ≤ 1/4, again by Proposition 3.

• Z 5,3 (x) = x(1 + 6x 2 + 6x 3 + x 5 ) = x(1 + x)(3x 2 + (1 + x 2 ) 2 -x(1 -x) 2 ), so that F α 5,3 = |Q α | 4 (Q 2 α + Q2 α )(3|Q α | 8 + (Q 4 α + Q4 α ) 2 -|Q α | 4 (Q 2 α -Q2 α ) 2 ). Since (Q 2 α -Q2 α ) 2 = -4(x -y) 2 sin 2 (πα), we see that F α 5,3 > 0 if α ≤ 1/2.
Observe that the generating function Z 5,3 has vanishing terms so that F α 5,3 cannot be suitably written in terms of F α 5,0 , F α 4,0 , F α 3,0 , F α 2,0 , contrary to F α 5,2 and F α 5,1 . In general, testing the positivity of F α n,k seems to depend on bizarre rearrangements.

Rectangular matrices.

In this paragraph we study in more details the minors A σ (1) and B α σ (1), with the notations of Section 3.3. Indeed, formulae (16), ( 17) and (18) show that the positivity of ∆ n α is ensured by that of A σ (1) and B α σ (1). The analysis for A σ (1) is easy.

Proposition 5. For every σ : {1, . . . , n} → {1, . . . , 2n -1} increasing, one has A σ (1) = A σ (1) > 0.

Proof. The generating function

r≥0 C r n-1+r x r = 1 (1 -x) n
and Edrei's criterion -see Theorem 1.2 p. 394 in [START_REF] Karlin | Total positivity[END_REF] -show that the sequence {C r n-1+r , r ≥ 0} is TP ∞ . In other words, the matrix A n (1) is TP ∞ viz. all its minors are non-negative. In particular, all coefficients A σ (1) are non-negative. We now compute its exact positive value with the help of a formula of Gessel and Viennot on binomial determinants. Substracting the i-th row from the (i -1)-th row successively for i = n . . . 2 and then repeating this operation for i = (n -1) . . . 2, i = (n -2) . . . 2, . . . we see indeed from the Pascal relationships C p+1 r = C p+1 r+1 -C p r that the minor A σ (1) is actually taken from the matrix

C k-i k-1 1≤i≤n,1≤k≤2n-1 = C i k 0≤i≤n-1,0≤k≤2n-2 .
The alternative formula mentioned at the bottom of p. 308 in [START_REF] Gessel | Binomial determinants, paths, and hook-length formulae[END_REF] entails

A σ (1) = 1≤i<j≤n (σ j -σ i ) sf(n -1) • In particular A σ (1) = A σ (1)
is always a positive integer (equal to one if and only if σ n -σ 1 = n -1).

The analysis for B α σ (1) is however much more delicate. After transposition, we see that it is the n × n minor obtained from the columns σ 1 , . . . , σ n in the horizontal matrix

C k-i n U α 2i-k 1≤i≤n,1≤k≤2n-1
.

Again, the generating function

  n j=i C j n C k-j n U α 2j-k if 1 ≤ i ≤ n -1, i ≤ k ≤ n + i, C k-n n U α 2n-k if i = n, n ≤ k ≤ 2n -1, 0 otherwise.
Observe that the (i, n + i) coefficient equals

n j=i C j n C n+i-j n U α 2j-n-i = [(n+i)/2] j=i C j n C n+i-j n (U α 2j-n-i + U α n+i-2j ) = 0,
so that we have a band-matrix of width n, whose (i, k) coefficient is given by

n j=i C j n C k-j n U α 2j-k if 1 ≤ i ≤ n, i ≤ k ≤ n -1 + i, 0 otherwise.
a criterion only on 2 × 2 minors, interestingly also related to the zeroes of Chebyshev polynomials of the second kind. Unfortunately, none of this criteria seems particularly helpful in our situation.

Proof of the corollary

Setting f α (x) = p α (1, x), we see from (3) that p α (t, x) = t -1/α f α (xt -1/α ), so that by Theorem 1.2.1. p.18 in [START_REF] Karlin | Total positivity[END_REF] the kernel p α (t, x) has the same sign-regularity over (0, +∞) × (0, +∞) than the kernel f α (e x-y ) over R × R. In Paragraph 7.12.E p.390 of [START_REF] Karlin | Total positivity[END_REF] it is shown that f α (e x-y ) ∈ STP ∞ ⇔ f α (e x-y ) ∈ TP ∞ ⇔ α ∈ {1/2, 1/3, . . . , 1/n, . . .} and the question is raised whether f α (e x-y ) should be TP of some finite order when α is not the reciprocal of an integer. In [START_REF] Simon | Multiplicative strong unimodality for positive stable laws[END_REF], we obtained the equivalence f α (e x-y ) ∈ TP 2 ⇔ α ≤ 1/2.

Let us now prove the Corollary. We need to show that if α is not the reciprocal of an integer and α < 1/n, then f α (e x-y ) ∈ SR n . If this were true, then the kernels f α (e y-x ) and e x-y R f α (e x-u )f α (e u-y ) du would also be SR n by Theorem 1.2.1. and Lemma 3.1.1. in [START_REF] Karlin | Total positivity[END_REF]. However, a well-known fractional moment identification -see (3.1) in [START_REF] Simon | Multiplicative strong unimodality for positive stable laws[END_REF] and the references therein -shows that the latter kernel equals g α (x -y) with the notations of Section 2. Hence, we get a contradiction to Part (b) of the theorem.

We finish this paper with a natural conjecture on the total positivity of the positive stable kernel p α , which reformulates Karlin's question in a more precise manner. By Lemma 3.1.1. in [START_REF] Karlin | Total positivity[END_REF] and the same argument as in the proof of the corollary, this would also show the conjecture stated in the introduction. But we believe that this last conjecture is harder because the kernel p α is not explicit in general. Acknowledgements. I am grateful to Arno Kuijlaars for having lent me his personal copy of [START_REF] Bressoud | Proofs and confirmations. The story of the alternating sign matrix conjecture[END_REF]. Ce travail a bénéficié d'une aide de l'Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.

Remark 1 .

 1 (a) The above argument based on the composition formula yields the STP ∞ character of all kernels (ax + by + c) -d for a, b, d > 0 and c ≥ 0 (this is the main result of [4] -see Theorem 3.1 therein), in writing 1 (ax + by + c) d = ∞ 0 e -axu e -byu u d-1 e -cu du.

  x r = (1 + x) n and Edrei's criterion show that the matrix C k-i n 1≤i≤n,1≤k≤2n-1 is TP ∞ . But since some U α 2i-k are negative, nothing can be said a priori about the non-negativity of B α σ (1). Transforming the rows (L 1 , . . . , L n ) through the simultaneaous linear operationsL i → n j=i C j n L jmultiplies the n × n minors by a constant positive factor and yields a matrix whose (i, k) coefficient is given by 

Conjecture 4 .

 4 For every n ≥ 2 one hasp α ∈ STP n ⇔ p α ∈ SR n ⇔ α ∈ {1/2, 1/3, . . . , 1/n, . . .} or α < 1/n.

We set B α,n for the above matrix, whose coefficients are all non-negative when α < 1/n. We also remark that B α,n is persymmetric, viz. B α,n (i, k) = B α,n (n + 1 -i, 2n -k) for any (i, k). Indeed,

and the last sum clearly vanishes. This persymmetry entails B σ (1) = B σ [START_REF] Behrend | Multiply-refined enumeration of alternating sign matrices[END_REF]. We now state a natural conjecture which would entail (24).

Conjecture 2. For every n ≥ 2, all coefficients B σ (1) are positive whenever α < 1/n.

The difficulty to prove this conjecture comes from the rectangular shape of the matrices {B α,n }, which makes it seemingly impossible to use any kind of induction argument. Let us finally display the five first elements of the sequence {B α,n }, whose common shape reminds that of Trinidad and Tobago's national flag (the red background being the zeroes, the crucial white diagonal stripes Chebyshev polynomials of the second kind, and the black central parallelogram positive linear combinations thereof) and displays some spatial unimodality around the middle.

Using some elementary trigonometry, for these first five values of n we could check that B α,n is TP (viz. all its minors are non-negative) if α < 1/n The following is hence natural Conjecture 3. For every n ≥ 1, the matrix B α,n is TP whenever α < 1/n. This conjecture is stronger than Conjecture 2 since it involves all minors. Notice that several criteria have appeared in recent years to prove the total positivity of a given matrix without checking every minor. See all the results mentioned in Section 2.5 of [START_REF] Pinkus | Totally positive matrices[END_REF] and also Theorem 2.16 therein for