In combinatorial mathematics, a t-core partition is a partition which has no hooks of length t. Such partitions have been used in the study of Ramanujan's congruences on the partition function^[1] and for representation theory of the symmetric group,^[2] especially modular representation theory.^[3]

A partition ${\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{k})}$ is a weakly decreasing list of positive integers, which we associate with its Young diagram by drawing ${\displaystyle \lambda _{i}}$ square cells in row i of an array. The size of a partition, denoted ${\displaystyle |\lambda |}$, is the sum of all ${\displaystyle \lambda _{i}}$, or the total number of cells. The conjugate partition ${\displaystyle \lambda '}$ is the partition whose values are the length of each column in ${\displaystyle \lambda }$.

The cells in the diagram are labelled by ${\displaystyle (i,j)}$ for ${\displaystyle 1\leq i\leq k}$ and ${\displaystyle 1\leq j\leq \lambda _{i}}$. The hook length ${\displaystyle h_{\lambda }(i,j)}$ of cell ${\displaystyle (i,j)}$ is given by $${\displaystyle h_{\lambda }(i,j)=\lambda _{i}+\lambda _{j}'-i-j+1,}$$ which is equal to the number of cells in the rotated L-shaped hook with vertex at ${\displaystyle (i,j)}$ which extends to the right and downwards.

For a positive integer t, a partition is t-core if it has no cells with a hook length of t.

If a partition is t-core, then it is also (nt)-core for every positive integer n.

In any row/column of a t-core partition which contains a cell of hook length h > t, there is a cell in the same row/column with hook length h – t.^[4]

The only 1-core partition is the empty partition with no cells. The 2-core partitions are the staircase partitions ${\displaystyle \lambda =(k,\dots ,2,1)}$. If ${\displaystyle \rho }$ is the perimeter of the Young diagram of a partition, then this partition is t-core for every ${\displaystyle t\geq \rho /2}$.

For every integer t at least 4, there exists a t-core partition of size n for every positive integer n. This result was known as the t-core conjecture before finally being proven by Andrew Granville and Ken Ono in 1996.^[5]

If ${\displaystyle c_{t}(n)}$ is the number of t-core partitions of size n, we have the generating function $${\displaystyle \sum _{n=0}^{\infty }c_{t}(n)q^{n}={\frac {\phi \left(q^{t}\right)^{t}}{\phi (q)}}}$$ where ${\displaystyle \phi (q)=(1-q)\left(1-q^{2}\right)\left(1-q^{3}\right)\cdots }$ is the Euler function. Note that ${\displaystyle 1/\phi (q)}$ is the generating function for all partitions.

The generating function for t-core partitions implies that there is a bijection between partitions ${\displaystyle \lambda }$ and pairs ${\textstyle \left(\lambda ^{(t)},\Lambda _{(t)}\right)}$, where ${\displaystyle \lambda ^{(t)}}$ is a t-core partition and ${\textstyle \Lambda _{(t)}=\left(\lambda _{(1)},\dots ,\lambda _{(t-1)}\right)}$ is a sequence of partitions such that $${\displaystyle |\lambda |=\left|\lambda ^{(t)}\right|+t\left(\left|\lambda _{(0)}\right|+\cdots +\left|\lambda _{(t-1)}\right|\right).}$$ We call ${\displaystyle \lambda ^{(t)}}$ the t-core of ${\displaystyle \lambda }$ and ${\displaystyle \Lambda ^{(t)}}$ the t-quotient of ${\displaystyle \lambda }$.

We give a construction of this bijection using an abacus.^[4] For a partition ${\displaystyle \lambda }$, define the infinite set ${\displaystyle \beta ^{\lambda }=\{\lambda _{i}-i\mid i\geq 1\}}$ where ${\displaystyle \lambda _{i}=0}$ for every ${\displaystyle i>k}$. Given the set ${\displaystyle \beta }$, we can recover ${\displaystyle \lambda }$ as follows: shift all entries of ${\displaystyle \beta }$ so that 0 is the smallest number which doesn't appear. Then the positive entries of this shifted set are the hook lengths of the first column of ${\displaystyle \lambda }$.

Consider an abacus with t infinitely long vertical runners numbered 0, 1, up to t – 1. Label the position on runner ${\displaystyle r}$ at height ${\displaystyle s}$ by ${\displaystyle st+r}$, so values increase left-to-right then bottom-to-top.

Given a partition ${\displaystyle \lambda }$, place beads on the abacus at each position in ${\displaystyle \beta ^{\lambda }}$. If ${\displaystyle \beta _{(r)}}$ are the heights of the beads on runner ${\displaystyle r}$, then ${\displaystyle \lambda _{(r)}}$ is the unique partition with ${\displaystyle \beta ^{\lambda _{(r)}}=\beta _{(r)}}$. Next, suppose ${\displaystyle \beta '}$ are the positions of the beads when the beads in ${\displaystyle \beta ^{\lambda }}$ naturally fall under gravity. Then ${\displaystyle \lambda ^{(t)}}$ is the unique partition satisfying ${\displaystyle \beta ^{\lambda ^{(t)}}=\beta '}$.

Suppose ${\displaystyle \lambda =(11,6,6,3,3,2,1,1)}$ and ${\displaystyle t=4}$. Then $${\displaystyle \beta ^{\lambda }=\{10,4,3,-1,-2,-4,-6,-7,-9,-10,-11,\dots \}}$$

We draw our 4-abacus by circling the beads in ${\displaystyle \beta ^{\lambda }}$.

Looking at runner 0 (the first column), the shaded beads have heights ${\displaystyle \beta _{(0)}=\{1,-1,-3,-4,-5,\dots \}}$. Hence, the first-column hook lengths of ${\displaystyle \lambda _{(1)}}$ are ${\displaystyle \{3,1\}}$, and so ${\displaystyle \lambda _{(0)}=(2,1).}$

In runner 1, we have ${\displaystyle \beta _{(1)}=\{-2,-3,\dots \}}$ and so ${\displaystyle \lambda _{(1)}}$ is the empty partition ${\displaystyle ()}$. We have ${\displaystyle \beta _{(2)}=\{2,-1,-2,\dots \}}$ so ${\displaystyle \lambda _{(2)}=(2)}$, and finally ${\displaystyle \lambda _{(3)}=(1,1)}$. These partitions make up the 4-quotient ${\displaystyle \Lambda ^{(4)}}$.

Now we calculate the 4-core of ${\displaystyle \lambda }$. Letting the beads of ${\displaystyle \beta ^{\lambda }}$ fall under gravity gives the abacus:

Therefore, $${\displaystyle \beta '=\{2,-1,-2,-4,-5,-6,\dots \}.}$$ The smallest missing value is –3, so shifting the values by 3 gives the first-column hook lengths ${\displaystyle \{5,2,1\}}$ which means ${\displaystyle \lambda ^{(4)}=(3,1,1)}$, which is indeed a 4-core partition.

Ramanujan's modular equations can be used to prove identities for ${\displaystyle c_{t}(n)}$, such as ${\displaystyle c_{3}(4n+1)=c_{3}(n)}$ and ${\displaystyle c_{5}(4n+3)=c_{5}(2n+1)+2c_{5}(n)}$.^[6]

Partitions which are simultaneously t-core for multiple values of t are well-studied.^[7] For example, if s and t are coprime positive integers, then the number of partitions which are simultaneously s-core and t-core is equal to $${\displaystyle {\frac {1}{s+t}}{\binom {s+t}{s}},}$$ which is a rational Catalan number.^[8]

The number of t-core partitions with at most k rows is equal to the number of partitions with at most k rows and at most t – 1 columns. A bijection between these sets is given by ${\displaystyle \lambda \mapsto \partial }$, where ${\displaystyle \partial _{i}}$ is the number of cells in row ${\displaystyle i}$ of ${\displaystyle \lambda }$ whose hook length is less than t.^[9]

• Hook length formula
• Representation theory of the symmetric group
• Crank of a partition