Partition of an integer ($\lambda ⊢n$)

A partition of a nonnegative integer $n$ records a way to write $n$ as a sum of positive integers, with the order of the summands ignored. Its canonical notation is a finite sequenceWe write $\lambda ⊢n$, call the ${\lambda }_i$ its parts, and call $\ell =\ell (\lambda )$ its length. Trailing zero parts may be appended for calculations without changing the partition of an integer; zero has the unique empty partition of an integer, of length zero. Thus $(3,2)$, $(3,2,0)$ and $5=2+3$ describe the same partition of an integer of five. The Young diagram below makes its two parts visible as rows of three and two cells.

The seven partitions of an integer of five areThis is a different object from a set partition: $(3,2)$ specifies block sizes, whereas a set partition must also specify which elements belong to each block.

Why this is useful: a partition of an integer is a compact description of a shape, yet it also labels an irreducible representation of a symmetric group over the complex numbers, realized by a Specht module. It therefore connects arithmetic decompositions, Young diagrams, and representation theory of the symmetric group. The generating function for partitions with bounded parts gives another way to count such shapes. Historically, Euler studied these counting problems using generating functions; his ^[ref] documents this early development.

Pad partitions with trailing zeros. In the descending-dictionary convention used for symmetric-function reverse lexicographic ordering, the partition with the larger first differing part precedes the other. This is the reverse of increasing dictionary order on integer partitions, and is a total order. It is distinct from comparing the last differing part in a monomial reverse-lexicographic order.

Fix a prime $p$. Replace each part $p^{a}m$, with $p\nmid m$, by $p^a$ copies of $m$. This is a size-preserving bijection from regular partitions to partitions of an integer whose parts are not divisible by $p$. To invert it, write each multiplicity of a part $m$ not divisible by $p$ in base $p$ and use its digit in position $a$ as the multiplicity of $p^{a}m$. Every resulting multiplicity is below $p$. The bijection explains why the number of simple symmetric-group modules equals the number of p-regular elements' conjugacy classes.

Pad partitions with trailing zeros and compare their first unequal parts. A partition is larger in dictionary order on integer partitions when that first unequal part is larger. This is a total order, whereas dominance order on partitions compares every prefix sum and is generally partial. The row-column collision lemma links these two orders.

The Young lattice is a partially ordered set whose elements are integer partitions, including the empty partition. Each partition of an integer is identified with its diagram, and the order is inclusion of Young diagrams. A cover adds exactly one box. A saturated path from the empty diagram to $\lambda$ is equivalent to a standard Young tableau of shape $\lambda$.

The Young branching graph is the graded cover graph of the Young lattice. Its level $n$ consists of partitions of $n$, and its edges add one box. It is also the branching graph for complex Specht modules under the restriction branching rule for a symmetric group.

An oscillating tableau is a sequence of Young diagrams, each obtained from its predecessor by adding or removing one box. These are paths in the Young branching graph with both directions allowed. A path of length $\ell$ from the empty diagram to $\lambda ⊢n$, with $\ell -n=2m\ge 0$, has count $\ell !f_{\lambda }/(2^{m}n!m!)$. This differs from a saturated upward path, whose length must equal $n$.

The generating function for partitions whose parts are at most $e$ isConjugating Young diagrams shows that it also counts partitions with at most $e$ parts. Multiplication by $x^e$ gives the generating function for partitions with exactly $e$ positive parts.

A self-conjugate partition is determined by the hook lengths of its diagonal cells. These are distinct odd positive integers, and every partition into distinct odd parts arises uniquely this way. Hence their generating function is

The Young diagram of a partition of an integer $\lambda$ consists of one unit square, or cell, for each pairThroughout this article we use the English convention: rows are left-aligned, row $1$ is at the top, row indices increase downwards, and column indices increase to the right. The row lengths weakly decrease from top to bottom. Thus $Y(3,2)$ has cells $(1,1),(1,2),(1,3),(2,1),(2,2)$, with three cells above two. In the French convention the longest row is drawn at the bottom; the underlying partition of an integer is unchanged, but directions in descriptions of hooks of a Young diagram and standard Young tableaux must be interpreted consistently. See Grinberg's hook-length-formula teaching slides, §§1.1–1.2, for diagrams with the same orientation.

The number of cells is $|\lambda |$, and reading the row lengths recovers the partition of an integer. Reading the column lengths instead gives its conjugate partition. This turns numerical inequalities into visible geometry: Young diagrams satisfywith missing parts treated as zero. For example, $(2,1)\subseteq (3,2)$. This inclusion is the order in Young's lattice; it differs from dominance order on partitions, which compares sums of initial parts of partitions of an integer of the same size. The partition of an integer $(3,2)$ dominates $(2,2,1)$ because their successive sums are $3\ge 2$, $5\ge 4$, and $5=5$, but neither Young diagram contains the other: one has cell $(1,3)$, the other $(3,1)$.

The beauty of a Young diagram is that its shape simultaneously records a partition of an integer, indexes a Specht module, and supports local quantities such as hook lengths that determine a global dimension. Filling its cells leads to Young tableaux; adding one cell at a time leads to the Young branching graph.

For diagram inclusion $\mu \subseteq \lambda$, the skew Young diagram $\lambda /\mu$ consists of the cells of $\lambda$ outside $\mu$. Diagram inclusion is not the dominance order on partitions. Edge adjacency defines its connected components, and a connected skew diagram without a $2\times 2$ square is a rim hook.

A standard skew Young tableau fills the $k$ cells of a skew Young diagram bijectively with $1,\dots ,k$, increasing along each row and column. Equivalently it is a linear extension of a partially ordered set formed from the row and column inequalities. Interchanging consecutive labels preserves standardness exactly when their cells are incomparable in that partial order.

Here $n=|\lambda |-|\mu |$, with zero entries for negative factorial arguments. Extract the squarefree monomial $x_1\cdots x_n$ from the skew Jacobi–Trudi identity. A product $h_{r_1}\cdots h_{r_{\ell }}$ contributes $n!/\prod r_i!$ because its distinct labels are assigned to the factors in that many ways. Summing determinant signs proves the formula, since a squarefree semistandard Young tableau is a standard skew Young tableau.

A horizontal strip is a skew Young diagram containing at most one cell in each column. Equivalently, $\lambda /\mu$ is a horizontal strip when ${\lambda }_i\ge {\mu }_i\ge {\lambda }_{i+1}$ for all $i$. The multiplicity of the trivial representation in the associated skew representation of a symmetric group is one for horizontal strips and zero otherwise. A column with two cells supplies a cyclic tableau vector on which an adjacent transposition acts by $-1$, excluding invariants.

Under edge adjacency, a skew Young diagram is totally disconnected when every component has one cell. This means that no two cells share an edge. It is stronger than being a horizontal strip, which excludes repeated columns but permits adjacent cells in one row. The trivial-constituent criterion for a skew representation of a symmetric group concerns horizontal strips, not total disconnection in this sense.

A hook partition has shape $(n-k,1^k)$, $0\le k\le n-1$. Equivalently its Young diagram has no cell $(2,2)$, so its only cell of content zero is $(1,1)$. Its standard Young tableaux are counted by $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$: choose the $k$ entries below the initial $1$, then the column and remaining row orders are forced.

An addable node is a cell outside a Young diagram whose insertion leaves a Young diagram. Its left and upper predecessors, when present in the positive-coordinate grid, must already belong to the diagram. Different addable nodes have distinct Young-diagram cell contents.

A removable node is a cell whose deletion leaves a Young diagram. The set ${\lambda }^{-}$ consists of the distinct partitions obtained by deleting one removable node from $\lambda$.

The conjugate partition exchanges rows and columns in a Young diagram: the cell $(i,j)$ becomes $(j,i)$. Equivalently,For $\lambda =(3,2)$, the three column lengths are $2,2,1$, so ${\lambda }^{\prime }=(2,2,1)$, as shown in Young diagram. Conjugating twice returns the original partition of an integer, and the number of cells stays five. Transposition also preserves hook lengths, with $h_{j,i}({\lambda }^{\prime })=h_{i,j}(\lambda )$; consequently the hook-length formula gives equal dimensions for the two Specht modules over the complex numbers. This row-column symmetry is already used in §1 of ^[ref].

In the English convention of Young diagram, the hook at a cell $(i,j)$ consists of that cell itself, all cells to its right in the same row, and all cells below it in the same column:Here ${\lambda }^{\prime }$ is the conjugate partition. The cells to the right form the arm of a Young-diagram cell, and those below form the leg of a Young-diagram cell. Both extend all the way to the edge of the Young diagram, rather than containing just the immediate neighbors. The three pieces are disjoint, so their total cardinality is one plus the two lengths.

For $(3,2)$, the highlighted Young-diagram hook is $H_{1,2}=\{(1,2),(1,3),(2,2)\}$, of hook length three. The hook at $(1,1)$ is $\{(1,1),(1,2),(1,3),(2,1)\}$, of hook length four: the southeast cell $(2,2)$ is excluded because it is in neither the anchor's row nor its column. These local shapes are useful because their hook lengths give the denominator of the hook-length formula. The hook terminology and its use to calculate representation degrees appear explicitly in §1 of ^[ref].

The arm consists of the cells strictly to the right of $(i,j)$ in its row of the Young diagram; its length is $a_{\lambda }(i,j)={\lambda }_i-j$. It excludes the anchor cell. At $(1,2)$ in $(3,2)$ the arm contains only $(1,3)$, so its length is one. Together with the leg of a Young-diagram cell and the anchor, it forms the hook of a Young diagram.

The leg consists of the cells strictly below $(i,j)$ in its column of the Young diagram; its length is $l_{\lambda }(i,j)={\lambda }_j^{\prime }-i$, where ${\lambda }^{\prime }$ is the conjugate partition. It excludes the anchor cell. At $(1,2)$ in $(3,2)$ the leg contains only $(2,2)$, so its length is one. Transposing the Young diagram exchanges the leg and the arm of a Young-diagram cell.

The hook length of cell $(i,j)$ in the Young diagram of a partition of an integer $\lambda$ is the number of cells in its Young-diagram hook:Here ${\lambda }^{\prime }$ is the conjugate partition. There is one hook length for every cell, so the multiset of hook lengths has cardinality $|\lambda |$.

For $\lambda =(3,2)$ and ${\lambda }^{\prime }=(2,2,1)$, compute $h_{1,1}=(3-1)+(2-1)+1=4$ and $h_{1,2}=(3-2)+(2-1)+1=3$. Continuing cell by cell gives the hook graph of a partitionor the row array $[[4,3,1],[2,1]]$. Its hook product is $4\cdot 3\cdot 1\cdot 2\cdot 1=24$. A cell has hook length one exactly when it is a Removable node of a Young diagram: it has no cell to its right or below. Here these are $(1,3)$ and $(2,2)$. Transposing the Young diagram gives the array $[[4,2],[3,1],[1]]$ with the same hook product; this explains geometrically why conjugate shapes have equal counts in the hook-length formula.

The hook graph is the Young diagram with the hook length written in each box. It displays the hook product and thus the hook-length formula for the corresponding Specht module. This use of the name is recorded in Frame, Robinson and Thrall's original hook-graph paper; it refers to a labeled array, not a directed hook-walk graph.

The product of all hook lengths in a Young diagram is the denominator in the hook-length formula, and the scalar in the quasi-idempotence identity for its Young symmetrizer.

For $(i,j)\in Y(\lambda )$, the integers from one through $h_{i,j}$ split as the disjoint unionTracing the southeast boundary of the hook records the first set at horizontal steps and the second at vertical steps.

If a partition has a hook of length $ef$, it has a hook of length $e$. In a beta set, the first hook is a bead-gap pair at distance $ef$; along the arithmetic progression with step $e$, some consecutive pair changes from a bead to a gap.

A rim hook, or border strip, is a connected skew Young diagram containing no $2\times 2$ square. Its leg length is one less than its number of occupied rows.

For a type $\alpha$, each successive difference is a rim hook of size ${\alpha }_i$; zero parts give empty differences of height zero. Label its cells by $i$. Such fillings are weakly increasing along rows and columns, and the height is the sum of the occupied strips' numbers of rows minus one. Empty parts are omitted from the associated product of positive power sums; the finite-variable degree-zero power sum is not inserted.

The content of the cell $(i,j)$ is $c_{i,j}=j-i$. Transposing the diagram negates every content and preserves every hook length.

The principal hook lengths are $h_{1,1},h_{2,2},\dots$ along the main diagonal. They are strictly decreasing positive integers, and their sum is the size of the partition.

For $m\ge \ell (\lambda )$, a beta set isA hook of length $h$ corresponds to a bead $b$ and an empty position $b-h$; removing the hook moves the bead to that gap.

For a partition padded to $l$ rows, its beta numbers are the distinct nonnegative integers ${\lambda }_i+l-i$. They are its occupied abacus of a partition positions. With exactly the positive row count, they equal the first-column hook lengths; extra zero rows change the representation of the beta set while preserving the partition. Individual beta numbers describe bead positions, whereas the beta set is the collection of all these numbers.

For ${\ell }_i={\lambda }_i+m-i$, the hook lengths in row $i$ are ${\ell }_i-q$, where $q$ runs over the non-beta numbers in $[0,{\ell }_i-1]$. Their product is ${\ell }_i!/{\prod }_{k>i}({\ell }_i-{\ell }_k)$. Multiplying over rows proves the identity and converts the hook-length formula into a Vandermonde determinant divided by factorials. Padding the partition with zero rows changes the beta set but not the hook product.

Repeated coordinates contribute zero. Sorting a distinct tuple and subtracting the staircase gives a partition of $m$, and its $m!$ orderings have identical squared summands. The beta-set hook-product identity turns each summand into $(\dim S^{\lambda }/m!{)}^2$. Summing and using the Artin–Wedderburn theorem for $\mathbb{C}{S}_m$ gives $m!{\sum }_{\lambda ⊢m}(\dim S^{\lambda }{)}^2/(m!{)}^2=1$. This is a normalized group algebra dimension identity.

Let $X=\{h_1,\dots ,h_m\}$ be a beta set and let $H_i(\lambda )$ be the hook lengths in row $i$. ThenThus hooks are exactly bead-gap pairs on the partition abacus.

An $e$-runner abacus arranges the nonnegative integers by their residues modulo $e$ and places beads at the positions in a beta set. Moving a bead up one place on its runner removes an $e$-hook.

An abacus represents the beta set of a partition by beads at integer positions. Positions with the same residue modulo the chosen runner count lie on one runner; moving a bead to an empty position above it represents the corresponding rim-hook removal.

Fix a positive integer $e$. A partition of an integer is uniquely specified by its core of a partition for modulus $e$ and its ordered quotient of a partition for that modulus. More precisely, for every $e$-core $\gamma$ and ordered tuple $({\nu }_0,\dots ,{\nu }_{e-1})$ of partitions, there is a unique partition $\lambda$ such thatThe ordered runners use the usual abacus of a partition convention with a number of beads divisible by $e$. Adding $e$ beads preserves both the tuple and its runner labels. The core fixes the relative runner offsets, and each component partition fixes the displacement of the beads on that runner, giving existence and uniqueness.

The $e$-core is obtained by repeatedly removing hooks of length $e$. On an $e$-runner abacus it is obtained by sliding every bead as high as possible, which proves independence of the order of removals.

The $e$-weight is the number of $e$-hooks removed to reach the $e$-core. It satisfiesand equals the total size of the partitions in the $e$-quotient.

Removing a rim hook of length two preserves the number of odd hook lengths minus the number of even hook lengths. If the 2-core is the staircase $(r,r-1,\dots ,1)$, every one of its $r(r+1)/2$ hooks is odd. Hence the difference for the original partition is

The $e$-quotient $Q_e(\lambda )=({\lambda }^{(0)},\dots ,{\lambda }^{(e-1)})$ is the tuple of partitions represented by the individual runners after their positions are divided by $e$.

Hooks of $\lambda$ whose lengths are divisible by $e$ correspond bijectively to hooks in the $e$-quotient. A bead-gap pair on one runner with distance $eh$ becomes a bead-gap pair of distance $h$ in that runner partition, and hook removal commutes with this correspondence.

The $e$-quotient tower recursively takes the $e$-quotient of every partition at the preceding level. For $e>1$, the sum of the sizes at each new level is at most $1/e$ times the preceding sum, so every partition has finite depth. For $e=1$, the quotient is the original partition and the tower has finite depth only for the empty partition.

With runners ordered by residues $0,1,2,3$, an abacus for $(3,1)$ gives

The nonempty levels of the 2-quotient tower of $(3,1)$ areand

For every $e,k\ge 1$, the $e^k$-quotient $Q_{e^k}(\lambda )$ is a permutation of level $k$ of the $e$-quotient tower. Writing a runner residue modulo $e^k$ in base $e$ shows that taking one quotient chooses one digit at a time; iteration may reverse the order of those digits but selects the same runner partitions.

The $p$-core tower places at level $r$ the $p$-cores of all partitions at level $r$ of the quotient tower of a partition. If $q_r$ is the sum of the sizes at quotient level $r$ and $c_r$ the sum at core level $r$, then

For a prime number $p$, integer $k\ge 1$, and $\gamma =C_{p^k}(\lambda )$, taking the $p^k$-core of a partition preserves the entries of the core tower of a partition at levels $0,\dots ,k-1$ and makes every level at least $k$ empty. Also the weight of a partition $w_{p^k}(\lambda )$ equals the total size at level $k$ of the quotient tower of a partition.

For the first claim, a removable rim hook of length $p^k$ is a bead move by $p^k$ on the abacus of a partition. It preserves the numbers of beads on the $p$ runners and hence the $p$-core. Under the abacus divisible-hook correspondence, it becomes removal of a $p^{k-1}$-hook in one quotient component; iteration preserves the cores at the next $k-1$ levels. After all $p^k$-hooks are removed, every component at quotient level $k$ is empty, so every higher core level is empty as well. The weight assertion follows from iterated quotient equals a power quotient up to permutation and the size formula for the quotient of a partition.

If $n=|\lambda |$ and $d_p(n)$ is its base-$p$ digit sum, thenThis combines the Hook-length formula, Legendre formula, the abacus divisible-hook correspondence, and the recurrence $q_r=c_r+p{q}_{r+1}$.

Let $p$ be a prime number, let $n={\sum }_r{\alpha }_r{p}^r$ with $0\le {\alpha }_r<p$, and let $c_r$ be the sum of partition sizes at level $r$ of the core tower of a partition $\lambda ⊢n$. ThenIndeed $n={\sum }_r{c}_r{p}^r$. Carrying $p$ units from position $r$ to position $r+1$ decreases ${\sum }_r{c}_r$ by $p-1$. Normalization to the base-$p$ digits therefore gives ${\sum }_r{c}_r\ge {\sum }_r{\alpha }_r$, with equality exactly when no carry is needed. The P-adic valuation of a symmetric-group character degree from the core tower is the difference divided by $p-1$.

For every partition $\lambda$,The core-tower formula reduces this to subadditivity of the base-$p$ digit sum across the sizes of the quotient partitions.

The $e$-residue of a cell $(i,j)$ is $j-i$ modulo $e$, and the $e$-content is the multiset of cell residues. Removing an $e$-hook removes exactly one cell of each residue. Among partitions of a fixed size, two partitions have the same $e$-content exactly when they have the same $e$-core.

A Young tableau is a filling of a Young diagram with entries, typically positive integers. The shape is the partition of an integer giving its row lengths. We draw Young tableaux in the English convention of Young diagram, with left-aligned rows arranged from top to bottom. A filling need not obey increasing conditions unless these are specified: standard Young tableaux use each of $1,\dots ,n$ exactly once and strictly increase along rows and down columns; Semistandard Young tableaux permit repetitions and require weakly increasing rows and strictly increasing columns.

In the constructions for a symmetric group involving Young symmetrizers, tabloids, and polytabloids, an unqualified numbered Young tableau means a bijective filling by $1,\dots ,n$, with no increasing condition imposed. This narrower convention is compatible with the broader use needed for Semistandard Young tableaux. Historically, Young's work connected these fillings to representation theory of the symmetric group; in particular, the equality between standard Young tableau counts and irreducible representation degrees over the complex numbers predates the modern hook-length formula. See Stanley's Algebraic Combinatorics, historical notes to Chapter 8.

For a Young tableau $t$, let $r_t$ sum its row permutations and $c_t$ be the signed sum of its column permutations in the group algebra of the symmetric group. The product $h_t=c_t{r}_t$ is a Young symmetrizer. The other order is another conventional realization. It satisfies $h_t^2=H_{\lambda }{h}_t$, where $H_{\lambda }$ is the hook product of a partition, so division by that scalar gives a primitive idempotent in characteristic zero. Acting on a tensor power constructs a Schur module.

If the rows of a Young tableau of shape $\lambda$ have no repeated intersection with the columns of one of shape $\mu$, then ${\sum }_{i\le k}{\lambda }_i\le {\sum }_{i\le k}{\mu }_i$ for every $k$. If also $\lambda \ge \mu$ in dictionary order on integer partitions, the shapes are equal. Saturation of the prefix bounds places one entry of each eligible row in each column, giving $u=rct$ with $r\in R(t)$ and $c\in C(t)$. A collision instead gives a transposition that makes row symmetrization followed by the relevant column antisymmetrization vanish.

Insert the successive letters of a permutation by row insertion to obtain an insertion tableau $P$, recording the insertion times in the new boxes of $Q$. This is a bijection between permutations and pairs of standard Young tableaux of a common shape. Reverse insertion removes the box marked by the latest recording time. The first row of $P$ has length equal to the longest increasing subsequence.

List each pair $(i,j)$ with multiplicity $a_{ij}$, sorted first by $i$ and then by $j$. Apply row insertion to the lower letters $j$, and record the upper letters $i$ in the new cells. The row insertion comparison is strictly greater than the carried entry. The result is a pair of Semistandard Young tableaux of the same shape. Reverse insertion removes the rightmost cell containing the largest recording entry, then replaces the rightmost strictly smaller entry in each preceding row. It reconstructs the ordered list, proving the bijection. The insertion and recording contents are respectively the column sums and row sums of $A$.

At one matrix entry $a$, let $\rho ,\mu ,\nu ,\lambda$ be the shapes of the northwest, northeast, southwest and southeast matrix prefixes under the RSK correspondence. The displayed row-length recurrence expresses merged insertion paths: the new entry supplies the first-row carry, and the overlap of the two incoming row extensions beyond the northwest shape is bumped into the next row. Zero boundary shapes determine the whole diagram. Transposing the matrix exchanges $\mu ,\nu$, leaving the rule invariant.

For a symmetric nonnegative integer matrix, the two neighboring prefix shapes along the diagonal are equal. In the RSK growth-diagram local rule, set $\mu =\nu$ and take alternating sums of row lengths. The two occurrences of the alternating sum of $\mu$ cancel, leaving $o(\lambda )=o(\rho )+a_{ii}$. Induction along the diagonal proves the formula. Each column contributes one to this alternating sum exactly when its length is odd.

For a finite $0$-$1$ matrix, list its occupied pairs by increasing top entry and decreasing bottom entry within a top-entry tie. Perform ordinary semistandard row insertion on the bottom entries and record top entries in the new cells. The result is a bijection with same-shape tableaux $T,U$ for which $T$ and $U^t$ are Semistandard Young tableaux. The types of $T$ and $U$ are respectively the column-sum and row-sum vectors of the matrix.

Insert $x$ into the first row of a Young tableau by replacing its leftmost entry strictly greater than $x$, and carry the displaced entry into the next row. If no entry is greater, append $x$ and stop. The procedure preserves a semistandard Young tableau; when all entries are distinct it preserves a near Young tableau. It underlies the Robinson–Schensted correspondence.

During row insertion into a near Young tableau, the successive carried entries strictly increase while the columns in which they are bumped weakly decrease. Every old cell keeps its entry or receives a smaller one. These facts follow from increasing rows and strictly increasing columns.

A near Young tableau fills a Young diagram with distinct entries from a totally ordered alphabet, increasing along rows and down columns. Its entries need not be the initial interval $1,\dots ,n$. It is the natural intermediate object for row insertion of a new entry.

A standard Young tableau of shape $\lambda ⊢n$ is a Young tableau that uses $1,\dots ,n$ exactly once and strictly increases left to right along every row and top to bottom down every column. Formally, a bijection $T:Y(\lambda )\to \{1,\dots ,n\}$ is standard whenwhenever the indicated cells belong to the Young diagram. We write $\mathrm{SYT}(\lambda )$ for the set of such fillings and $f^{\lambda }=|\mathrm{SYT}(\lambda )|$ for its cardinality.

For $(3,2)$, the five standard Young tableaux are listed below, with each pair giving the top row followed by the bottom row:There are no others: the upper-left entry must be $1$, and choosing the two bottom-row entries fixes both increasing row orders. Of the six choices of two entries from $\{2,3,4,5\}$, only bottom row $(2,3)$ fails, because its $3$ would lie below $4$. Hence $f^{(3,2)}=5$ by direct enumeration, independently of the hook-length formula.

The cells occupied by $1,\dots ,r$ form a Young diagram: every cell above or to the left of an occupied cell has a smaller label. Consequently a standard Young tableau is equivalently a path from the empty shape to $\lambda$ in the Young branching graph, adjoining one addable node of a Young diagram at each step. For $T_2$, the successive shapes areConversely, number the cell added at step $r$ by $r$ to recover the standard Young tableau. This path interpretation also makes a standard Young tableau a linear extension of a partially ordered set defined by the row and column requirements; see Stanley's Algebraic Combinatorics, Chapter 8, on tableaux and paths in Young's lattice.

The usefulness goes beyond counting: the standard polytabloid basis of a Specht module $S^{\lambda }$ over the complex numbers is indexed by $\mathrm{SYT}(\lambda )$, so ${\dim }_{\mathbb{C}}{S}^{\lambda }=f^{\lambda }$. Each of the five pictured standard Young tableaux therefore labels one basis element of the irreducible representation $S^{(3,2)}$ of the symmetric group $S_5$, whose dimension over the complex numbers is five. The filled diagrams index the basis; the actual vectors are polytabloids. The hook-length formula computes this dimension without listing all the standard Young tableaux.

Read each column from top to bottom, with columns ordered from left to right, and compare the resulting words lexicographically. With Young symmetrizer convention $h_t=c_t{r}_t$, this order gives triangular vanishing of Young-symmetrizer products. Changing the symmetrizer product convention requires a compatible order, rather than retaining a fixed vanishing direction automatically.

For standard tableaux of the same shape and $h_t=c_t{r}_t$, $t>u$ in column-reading order of standard Young tableaux implies $h_t{h}_u=0$. If there were no row-column collision, each column of $u$ would select one entry from each eligible row of $t$. Its sorted first column is componentwise at least the row minima of $t$; equality fixes those selected entries and permits induction after deleting the column. Thus no collision implies $u\ge t$. A collision gives $r_t{c}_u=0$.

Swapping consecutive labels $i,i+1$ preserves a standard Young tableau exactly when their cells are incomparable in the row-and-column order. Their contents then differ by more than one in absolute value. These admissible swaps connect all standard tableaux of any fixed shape, as adjacent swaps connect linear extensions of a partially ordered set.

The content vector of a standard Young tableau $T$ is $(c_T(1),\dots ,c_T(n))$, where $c_T(r)$ is the Content of a Young-diagram cell containing $r$. Such vectors are exactly the integer vectors satisfying: the first coordinate is zero; each subsequent coordinate has an earlier neighbor differing by one; between two occurrences of $a$ both $a-1$ and $a+1$ occur. To reconstruct the tableau, insert each entry in the next available cell on its prescribed diagonal. The neighbor conditions supply its required predecessors, so each insertion is an addable node of a Young diagram.

The axial distance from label $j$ to label $i$ in a standard Young tableau is the difference of their Young-diagram cell contents. This sign convention makes $c_T(r+1)-c_T(r)$ equal to $+1$ for consecutive entries in a row and $-1$ in a column. The reciprocal distance supplies the diagonal coefficient of Young orthogonal form.

A semistandard Young tableau fills a Young diagram with positive integers, permitting repeated entries, so that entries weakly increase along rows and strictly increase down columns. Its content records the number of occurrences of each entry. The Kostka number $K_{\lambda \mu }$ counts such fillings of shape $\lambda$ and content $\mu$.

For adjacent letters $i,i+1$, pair entries in the same column and leave those pairs fixed. In each row, interchange the numbers of unpaired $i$ and $i+1$ entries, preserving their weak order. Strict columns are preserved: a letter that could obstruct a change from $i$ to $i+1$, or conversely, would already have been paired. Applying the operation twice restores the tableau. The paired letters contribute equal amounts to the two contents, so the involution swaps their total multiplicities.

A word in positive integers is a lattice word if every initial segment has at least as many $i$'s as $(i+1)$'s for every $i$. This is the reading-word condition in the Littlewood–Richardson rule. Under good-letter matching in a tableau word, every letter of a lattice word is good.

The row stabilizer $R(t)$ permutes entries within each row, while the column stabilizer $C(t)$ permutes entries within each column. Their intersection is trivial.

A tabloid is an equivalence class of tableaux under permutations within rows. The tabloids of shape $\lambda$ form a transitive $S_n$-set.

The Young permutation module is the permutation module on the $\lambda$-tabloids.

A tabloid is fixed by a permutation exactly when each of its cycles lies wholly within one row. Assigning the distinct length-$q$ cycles to rows gives the displayed coefficient. More explicitly, sum ${\prod }_q{m}_q!/{\prod }_j{a}_{qj}!$ over nonnegative integers $a_{qj}$ with ${\sum }_j{a}_{qj}=m_q$ and ${\sum }_{q}q{a}_{qj}={\lambda }_j$. Equal-length rows remain distinguished. Thus the character counts fixed ordered row sets, rather than unordered set partitions.

Over the complex numbers, the signed Young permutation module isThe isomorphism is the tensor identity for induced representations. The partition index is unchanged by this twist; conjugation instead changes the labels of its irreducible constituents. For the conjugate partition ${\lambda }^{\prime }$, the character inner product of $M^{\lambda }$ and ${\tilde{M}}^{{\lambda }^{\prime }}$ is one: Mackey restriction formula reduces it to the unique zero-one matrix with row margins $\lambda$ and column margins ${\lambda }^{\prime }$.

Over the complex numbers,where the Kostka number $K_{\alpha \lambda }$ counts semistandard tableaux of shape $\alpha$ and content $\lambda$. It is nonzero only when $\alpha$ dominates $\lambda$.

Let $M(\mu ,\lambda )$ be the multiplicity of $V^{\mu }$ in the complex Young permutation module $M^{\lambda }$. For $\lambda ⊢n$ and $\rho ⊢n-1$,Here $\nearrow$ means adding one cell, and zero parts are omitted. Equal row lengths contribute separately on the right. Restrict $M^{\lambda }$ to $S_{n-1}$ and separate the tabloid orbits by the row containing $n$ to obtain the right side. Decompose into irreducibles and use the restriction branching rule for a symmetric group to obtain the left side. Equating multiplicities proves the formula.

For $0\le k\le n/2$, over the complex numbers,By Young's rule, the multiplicity is the Kostka number for content $(n-k,k)$. A semistandard Young tableau on just the entries $1,2$ has at most two rows. For shape $(n-j,j)$, every lower entry is $2$ and every entry above it is $1$; the remaining top row is uniquely fixed by the content. Such a filling exists precisely for $j\le k$. Thus each displayed Specht module occurs once. A zero second part is omitted.

The Kostka number $K_{\alpha \lambda }$ is the number of semistandard Young tableaux of shape $\alpha$ and content $\lambda$: rows are weakly increasing and columns are strictly increasing.

For $n\ge 4$, identify $M^{(n-2,1^2)}$ with the permutation module on ordered pairs $(i,j)$ with $i\ne j$. Let $p_1(i,j)=e_i$, $p_2(i,j)=e_j$, and $q(i,j)=\{i,j\}$. With $U=\ker p_1$ and $V=U\cap \ker p_2$, one has a Specht filtrationwhose successive quotients areThe same construction for $n=3$ omits the zero $S^{(n-2,2)}$ factor.