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Kimberly, Alabama

Kimberly is a city in Jefferson County, Alabama, United States. At the 2020 census, the population was 3,841.

History
The city of Kimberly developed around a stagecoach station built in the 1830s; it remained in service until 1865. At this time, the settlement was known as Kennelsville. The city then became known as Jefferson, probably after the nearby Old Jefferson coal mine, until the late 1880s. In 1888, the town's post office was relocated to nearby Morris. In 1905, the town voted to change its name to Kimberly because another Alabama town had already claimed the name Jefferson. The town had a population of around 900 by 1910. Local coal mines were primary employers in Kimberly and were active through the 1940s, gradually closing down during the 1950s. Another significant employer in Kimberly was the Dixie Firebrick Company, founded in the 1930s; it was later bought out by A.P. Green Refractories Company. This facility shut down sometime in the mid-1970s. The city officially incorporated in 1952. Many city records and history were lost in a fire at the city hall in 1989. Beginning June 29, 2011, due to the population increasing from 1,801 persons in 2000 to 2,711 in 2010, the town began operating as a city, per Alabama law.

Geography
Kimberly is located at 33.77111°N, -86.79528°W (33.771163, -86.795280). According to the U.S. Census Bureau, the city has a total area of 4.0 sqmi, all land.

2020 census
As of the 2020 United States census, there were 3,841 people, 1,168 households, and 947 families residing in the city.

2010 census
At the 2010 census there were 2,711 people, 932 households, and 752 families living in the city. The population density was 677.5 PD/sqmi. There were 1,002 housing units at an average density of 250.5 /sqmi. The racial makeup of the city was 96.2% White, 1.8% Black or African American, 0.4% Native American, 0.6% Asian, 0.3% from other races, and .7% from two or more races. .8% of the population were Hispanic or Latino of any race. Of the 932 households 41.7% had children under the age of 18 living with them, 67.0% were married couples living together, 10.2% had a female householder with no husband present, and 19.3% were non-families. 16.8% of households were one person and 6.8% were one person aged 65 or older. The average household size was 2.91 and the average family size was 3.28.

The age distribution was 28.7% under the age of 18, 8.0% from 18 to 24, 29.0% from 25 to 44, 25.4% from 45 to 64, and 9.0% 65 or older. The median age was 35.4 years. For every 100 females, there were 94.9 males. For every 100 females age 18 and over, there were 96.9 males.

2000 census
At the 2000 census there were 1,801 people, 652 households, and 528 families living in the town. The population density was 447.8 PD/sqmi. There were 699 housing units at an average density of 173.8 /sqmi. The racial makeup of the town was 95.78% White, 1.61% Black or African American, 0.61% Native American, 0.17% Asian, 0.44% from other races, and 1.39% from two or more races. 1.17% of the population were Hispanic or Latino of any race. Of the 652 households 42.2% had children under the age of 18 living with them, 68.3% were married couples living together, 9.5% had a female householder with no husband present, and 18.9% were non-families. 17.0% of households were one person and 8.4% were one person aged 65 or older. The average household size was 2.76 and the average family size was 3.11.

The age distribution was 29.3% under the age of 18, 6.7% from 18 to 24, 33.1% from 25 to 44, 20.5% from 45 to 64, and 10.5% 65 or older. The median age was 34 years. For every 100 females, there were 99.4 males. For every 100 females age 18 and over, there were 91.6 males.

The median household income was $46,343 and the median family income was $52,109. Males had a median income of $36,977 versus $29,150 for females. The per capita income for the town was $17,055. About 6.7% of families and 7.7% of the population were below the poverty line, including 9.6% of those under age 18 and 15.8% of those age 65 or over.

Government

 * Elected Officials

 * Mayor - Bob Ellerbrock
 * Council Member/Mayor Pro Tem - Tanya Cowart
 * Council Member - Kevin Powell
 * Council Member-James Harris
 * Council Member - John Richardson
 * Council Member - Jay Jenkins

 * Department Heads

 * City Clerk - Sandy Waid
 * Police Chief - Ricky Pridmore
 * Fire Chief - Brendt Wood
 * Streets and Parks Director -

Education

 * Bryan Elementary School
 * North Jefferson Middle School
 * Mortimer Jordan High School

All three schools are located in Kimberly, with North Jefferson Middle and Mortimer Jordan serving a feeder pattern consisting of Kimberly, Warrior, Morris, Trafford and several areas of extreme north Gardendale.

title: Kimberly, Alabama


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Integer partition

In number theory and combinatorics, a partition of a non-negative integer $n$, also called an integer partition, is a way of writing $k$ as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, $4$ can be partitioned in five distinct ways:

The only partition of zero is the empty sum, having no parts.

The order-dependent composition $4$ is the same partition as $3 + 1$, and the two distinct compositions $2 + 2$ and $2 + 1 + 1$ represent the same partition as $1 + 1 + 1 + 1$.

An individual summand in a partition is called a part. The number of partitions of $n$ is given by the partition function $1 + 3$. So $3 + 1$. The notation $1 + 2 + 1$ means that $n$ is a partition of $n$.

Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation theory in general.

Examples
The seven partitions of 5 are
 * 5
 * 4 + 1
 * 3 + 2
 * 3 + 1 + 1
 * 2 + 2 + 1
 * 2 + 1 + 1 + 1
 * 1 + 1 + 1 + 1 + 1

Some authors treat a partition as a decreasing sequence of summands, rather than an expression with plus signs. For example, the partition 2 + 2 + 1 might instead be written as the tuple $1 + 1 + 2$ or in the even more compact form $2 + 1 + 1$ where the superscript indicates the number of repetitions of a part.

This multiplicity notation for a partition can be written alternatively as $$1^{m_1}2^{m_2}3^{m_3}\cdots$$, where $p(n)$ is the number of 1's, $p(4) = 5$ is the number of 2's, etc. (Components with $λ ⊢ n$ may be omitted.) For example, in this notation, the partitions of 5 are written $$5^1, 1^1 4^1, 2^1 3^1, 1^2 3^1, 1^1 2^2, 1^3 2^1$$, and $$1^5$$.

Diagrammatic representations of partitions
There are two common diagrammatic methods to represent partitions: as Ferrers diagrams, named after Norman Macleod Ferrers, and as Young diagrams, named after Alfred Young. Both have several possible conventions; here, we use English notation, with diagrams aligned in the upper-left corner.

Ferrers diagram
The partition 6 + 4 + 3 + 1 of the number 14 can be represented by the following diagram:

The 14 circles are lined up in 4 rows, each having the size of a part of the partition. The diagrams for the 5 partitions of the number 4 are shown below:

Young diagram
An alternative visual representation of an integer partition is its Young diagram (often also called a Ferrers diagram). Rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. Thus, the Young diagram for the partition 5 + 4 + 1 is
 * [[Image:Young diagram for 541 partition.svg|100px]]

while the Ferrers diagram for the same partition is

 * - style="vertical-align:top; text-align:left;"
 * GrayDot.svgGrayDot.svgGrayDot.svgGrayDot.svgGrayDot.svg
 * - style="vertical-align:top; text-align:center;"
 * }

While this seemingly trivial variation does not appear worthy of separate mention, Young diagrams turn out to be extremely useful in the study of symmetric functions and group representation theory: filling the boxes of Young diagrams with numbers (or sometimes more complicated objects) obeying various rules leads to a family of objects called Young tableaux, and these tableaux have combinatorial and representation-theoretic significance. As a type of shape made by adjacent squares joined together, Young diagrams are a special kind of polyomino.

Partition function

The partition function $$p(n)$$ counts the partitions of a non-negative integer $$n$$. For instance, $$p(4)=5$$ because the integer $$4$$ has the five partitions $$1+1+1+1$$, $$1+1+2$$, $$1+3$$, $$2+2$$, and $$4$$. The values of this function for $$n=0,1,2,\dots$$ are:
 * 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, ....

The generating function of $$p$$ is
 * $$\sum_{n=0}^{\infty}p(n)q^n=\prod_{j=1}^{\infty}\sum_{i=0}^{\infty}q^{ji}=\prod_{j=1}^{\infty}(1-q^j)^{-1}.$$

No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument., as follows:

 * $$p(n) \sim \frac {1} {4n\sqrt3} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right)$$ as $$n \to \infty$$

In 1937, Hans Rademacher found a way to represent the partition function $$p(n)$$ by the convergent series

$$p(n) = \frac{1}{\pi \sqrt{2}} \sum_{k=1}^\infty A_k(n)\sqrt{k} \cdot \frac{d}{dn} \left({   \frac {1} {\sqrt{n-\frac{1}{24}}}    \sinh \left[ {\frac{\pi}{k}    \sqrt{\frac{2}{3}\left(n-\frac{1}{24}\right)}}\,\,\,\right] }\right)$$ where

$$A_k(n) = \sum_{0 \le m < k, \; (m, k) = 1} e^{ \pi i \left( s(m, k) - 2 nm/k \right) }.$$ and $$s(m,k)$$ is the Dedekind sum.

The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument.

 * $$p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\cdots$$

Srinivasa Ramanujan discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of $$n$$ ends in the digit 4 or 9, the number of partitions of $$n$$ will be divisible by 5.

Restricted partitions
In both combinatorics and number theory, families of partitions subject to various restrictions are often studied. This section surveys a few such restrictions.

Conjugate and self-conjugate partitions
If we flip the diagram of the partition 6 + 4 + 3 + 1 along its main diagonal, we obtain another partition of 14:

By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 + 1 + 1 of the number 14. Such partitions are said to be conjugate of one another. In the case of the number 4, partitions 4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1 are conjugate of each other. Of particular interest are partitions, such as 2 + 2, which have themselves as conjugate. Such partitions are said to be self-conjugate.

Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.

Proof (outline): The crucial observation is that every odd part can be "folded" in the middle to form a self-conjugate diagram:

One can then obtain a bijection between the set of partitions with distinct odd parts and the set of self-conjugate partitions, as illustrated by the following example:

Odd parts and distinct parts
Among the 22 partitions of the number 8, there are 6 that contain only odd parts:
 * 7 + 1
 * 5 + 3
 * 5 + 1 + 1 + 1
 * 3 + 3 + 1 + 1
 * 3 + 1 + 1 + 1 + 1 + 1
 * 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Alternatively, we could count partitions in which no number occurs more than once. Such a partition is called a partition with distinct parts. If we count the partitions of 8 with distinct parts, we also obtain 6:
 * 8
 * 7 + 1
 * 6 + 2
 * 5 + 3
 * 5 + 2 + 1
 * 4 + 3 + 1

This is a general property. For each positive number, the number of partitions with odd parts equals the number of partitions with distinct parts, denoted by q(n). This result was proved by Leonhard Euler in 1748 and later was generalized as Glaisher's theorem.

For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction. An important example is q(n) (partitions into distinct parts). The first few values of q(n) are (starting with q(0)=1):
 * 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ....

The generating function for q(n) is given by
 * $$\sum_{n=0}^\infty q(n)x^n = \prod_{k=1}^\infty (1+x^k) = \prod_{k=1}^\infty \frac {1}{1-x^{2k-1}} .$$

The pentagonal number theorem gives a recurrence for q:
 * q(k) = ak + q(k − 1) + q(k − 2) − q(k − 5) − q(k − 7) + q(k − 12) + q(k − 15) − q(k − 22) − ...

where ak is (−1)m if k = 3m2 − m for some integer m and is 0 otherwise.

Restricted part size or number of parts
By taking conjugates, the number $(2, 2, 1)$ of partitions of $(2^{2}, 1)$ into exactly k parts is equal to the number of partitions of $m_{1}$ in which the largest part has size $m_{2}$. The function $m_{i} = 0$ satisfies the recurrence

with initial values $p_{k}(n)$ and $n$ if $n$ and $k$ and $p_{k}(n)$ are not both zero.

One recovers the function p(n) by

p(n) = \sum_{k = 0}^n p_k(n). $$

One possible generating function for such partitions, taking k fixed and n variable, is
 * $$ \sum_{n \geq 0} p_k(n) x^n = x^k\prod_{i = 1}^k \frac{1}{1 - x^i}.$$

More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function
 * $$\prod_{t \in T}(1-x^t)^{-1}.$$

This can be used to solve change-making problems (where the set T specifies the available coins). As two particular cases, one has that the number of partitions of n in which all parts are 1 or 2 (or, equivalently, the number of partitions of n into 1 or 2 parts) is
 * $$\left \lfloor \frac{n}{2}+1 \right \rfloor ,$$

and the number of partitions of n in which all parts are 1, 2 or 3 (or, equivalently, the number of partitions of n into at most three parts) is the nearest integer to (n + 3)2 / 12.

Partitions in a rectangle and Gaussian binomial coefficients
One may also simultaneously limit the number and size of the parts. Let $p_{k}(n) = p_{k}(n − k) + p_{k−1}(n − 1)$ denote the number of partitions of $λ$ with at most $n$ parts, each of size at most $n$. Equivalently, these are the partitions whose Young diagram fits inside an $p_{0}(0) = 1$ rectangle. There is a recurrence relation $$p(N,M;n) = p(N,M-1;n) + p(N-1,M;n-M)$$ obtained by observing that $$p(N,M;n) - p(N,M-1;n)$$ counts the partitions of $M$ into exactly $N$ parts of size at most $n$, and subtracting 1 from each part of such a partition yields a partition of $p_{k}(n) = 0$ into at most $M$ parts.

The Gaussian binomial coefficient is defined as: $${k+\ell \choose \ell}_q = {k+\ell \choose k}_q = \frac{\prod^{k+\ell}_{j=1}(1-q^j)}{\prod^{k}_{j=1}(1-q^j)\prod^{\ell}_{j=1}(1-q^j)}.$$ The Gaussian binomial coefficient is related to the generating function of $n ≤ 0 or k ≤ 0$ by the equality $$\sum^{MN}_{n=0}p(N,M;n)q^n = {M+N \choose M}_q.$$

Rank and Durfee square
The rank of a partition is the largest number k such that the partition contains at least k parts of size at least k. For example, the partition 4 + 3 + 3 + 2 + 1 + 1 has rank 3 because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. In the Ferrers diagram or Young diagram of a partition of rank r, the r × r square of entries in the upper-left is known as the Durfee square:

 * - style="vertical-align:top; text-align:left;"
 * RedDot.svgRedDot.svgRedDot.svgGrayDot.svg
 * }

The Durfee square has applications within combinatorics in the proofs of various partition identities. It also has some practical significance in the form of the h-index.

A different statistic is also sometimes called the rank of a partition (or Dyson rank), namely, the difference $$\lambda_k - k$$ for a partition of k parts with largest part $$\lambda_k$$. This statistic (which is unrelated to the one described above) appears in the study of Ramanujan congruences.

Young's lattice
There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set is known as Young's lattice. The lattice was originally defined in the context of representation theory, where it is used to describe the irreducible representations of symmetric groups Sn for all n, together with their branching properties, in characteristic zero. It also has received significant study for its purely combinatorial properties; notably, it is the motivating example of a differential poset.

Random partitions
There is a deep theory of random partitions chosen according to the uniform probability distribution on the symmetric group via the Robinson–Schensted correspondence. In 1977, Logan and Shepp, as well as Vershik and Kerov, showed that the Young diagram of a typical large partition becomes asympototically close to the graph of a certain analytic function minimizing a certain functional. In 1988, Baik, Deift and Johansson extended these results to determine the distribution of the longest increasing subsequence of a random permutation in terms of the Tracy–Widom distribution. Okounkov related these results to the combinatorics of Riemann surfaces and representation theory.

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title: Integer partition