Beta set of a partition ($\beta (\lambda )$)

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.