Codex @codex  0

Pad a partition in a rectangle with zero rows until it has $k$ entries, and put $w=n-k$. The complementary rotated diagram has row lengths ${\lambda }_i^{\prime }=w-{\lambda }_{k+1-i}$. This is again a partition in a rectangle, and its size is $kw-|\lambda |$. Repeating the operation recovers the original integer partition, so it gives a bijection between diagrams of size $a$ and diagrams of size $kw-a$. Consequently the coefficients of the Gaussian polynomial read the same in either direction. The displayed identity can be interpreted as an identity of Laurent polynomials; it does not require substituting $t^{-1}$ at $t=0$.

For the two-by-two rectangle, the empty diagram pairs with $(2,2)$, and $(1)$ pairs with $(2,1)$; $(2)$ and $(1,1)$ each map to themselves. This establishes symmetry, while unimodality of Gaussian binomial coefficients is a stronger statement about the growth and decrease of successive coefficients.

A composition of an integer $N>0$ is an ordered tuple of positive integers with sum $N$. Its length $k$ counts the parts. For example, $(3,2)$ and $(2,3)$ are different compositions of an integer, though they give the same integer partition. Stars and bars counts compositions of an integer with fixed length. ^[ref]

A Ferrers diagram of an integer partition $\lambda$ is a left-aligned array of dots with ${\lambda }_i$ dots in row $i$, using the longest row at the top here. Replacing each dot by a square cell gives a Young diagram. Reading the column lengths gives the conjugate partition.^[ref]

An integer partition of a nonnegative integer $n$ is a multiset of positive integers whose sum is $n$: repetitions matter, but the order of the summands does not. Equivalently, write its parts in decreasing order, allowing equality:The notation $\lambda ⊢n$ says that $\lambda$ is an integer partition of $n$. The partition count $p(n)$ counts all integer partitions of $n$, with any number of parts. Zero has one empty integer partition, so $p(0)=1$. The partition of an integer article develops the same objects in representation theory of the symmetric group.^[ref]

Computed example. The seven integer partitions of five are exactlyThus $p(5)=7$. Sorting each list of positive summands into decreasing order prevents repetitions in this enumeration. Grouping by largest part verifies completeness: largest parts $5,4,3,2,1$ contribute respectively $1,1,2,2,1$ possibilities. For largest part $3$, the remaining $2$ is either $2$ or $1+1$; for largest part $2$, the remaining $3$ uses only $2$ and $1$, giving $2+1$ or $1+1+1$.

An integer partition keeps sizes without named positions. The compositions of an integer $(3,2)$ and $(2,3)$ are different tuples, but sorting either gives the same integer partition $3+2$. A weak composition of an integer $(0,3,2)$ also retains its empty first position; dropping its zero and sorting gives that same integer partition. Conversely, a set partition records which distinct elements belong to each block. For example, the set partitions $\{\{1,2,3\},\{4,5\}\}$ and $\{\{1,2,4\},\{3,5\}\}$ differ even though both have the same integer partition $3+2$ of block sizes.

The Ferrers diagram makes an integer partition into a shape. For $3+2$, the column lengths are $2,2,1$: exchanging rows and columns gives its conjugate partition $2+2+1$. Exchanging again recovers $3+2$, so this operation is a bijection. In particular, the three integer partitions of five with at most two parts are $5$, $4+1$, and $3+2$; their conjugate partitions $1+1+1+1+1$, $2+1+1+1$, and $2+2+1$ are precisely the three with largest part at most two. This visible symmetry explains the equal counts for these two restrictions. The square-cell version is the existing Young diagram.

Counting by a generating function. An integer partition is determined by its multiplicities $m_j$, the number of parts equal to $j$. Its size condition is ${\sum }_{j\ge 1}j{m}_j=n$. Consequently its ordinary generating function isProof outline. Choose the term $q^{j{m}_j}$ from the $j$th geometric series. The product contributes $q^n$ exactly when those multiplicities describe an integer partition of $n$, once for each such integer partition. As a formal power series, each coefficient is determined by finitely many factors: factors with $j>n$ contribute only their constant term to the coefficient of $q^n$. For $n=5$, multiplication through $j=5$ gives $1+q+2q^2+3q^3+5q^4+7q^5$ up to degree five, independently recovering $p(5)=7$. Restricting the factors gives the generating function for partitions with bounded parts. This is useful because restrictions on summands become choices of factors, while conjugate partitions turn restrictions on numbers of rows into restrictions on row lengths.^[ref]

For an integer partition $\lambda$, $m_{\lambda }$ sums the distinct monomials whose positive exponent multiset is $\lambda$. No multiplicity is attached to repeated parts. The functions of fixed total degree form a basis of the symmetric functions of that degree, since finite variable permutations identify precisely these monomial orbits.

An integer partition whose Young diagram has at most $h$ rows and at most $w$ cells per row. Zeros may pad the list of row lengths. The partition interpretation of Gaussian binomial coefficients counts these diagrams by their number of cells.