Core partition

A 6-core partition, with the hook length of each cell labelled.

In combinatorial mathematics, a Template:Mvar-core partition is a partition which has no hooks of length Template:Mvar. Such partitions have been used in the study of Ramanujan's congruences on the partition function and for representation theory of the symmetric group, especially modular representation theory.

Definition

A partition $\lambda =({\lambda }_1,\dots ,{\lambda }_k)$ is a weakly decreasing list of positive integers, which we associate with its Young diagram by drawing ${\lambda }_i$ square cells in row Template:Mvar of an array. The size of a partition, denoted $|\lambda |$, is the sum of all ${\lambda }_i$, or the total number of cells. The conjugate partition ${\lambda }^{\prime }$ is the partition whose values are the length of each column in $\lambda$.

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

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

Properties

If a partition is Template:Mvar-core, then it is also (Template:Mvar)-core for every positive integer Template:Mvar.

In any row/column of a Template:Mvar-core partition which contains a cell of hook length Template:Mvar > Template:Mvar, there is a cell in the same row/column with hook length Template:Mvar – Template:Mvar.

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

Number of Template:Mvar-core partitions of size Template:Mvar.

Template:Diagonal split header	1	2	3	4	5	6
1	0	0	0	0	0	0
2	1	0	1	0	0	1
3	1	2	0	2	1	2
4	1	2	3	1	3	3
5	1	2	3	5	2	6
6	1	2	3	5	7	5

For every integer Template:Mvar at least Template:Math, there exists a Template:Mvar-core partition of size Template:Mvar for every positive integer Template:Mvar. This result was known as the Template:Mvar-core conjecture before finally being proven by Andrew Granville and Ken Ono in 1996.

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

Abacus

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

Construction

We give a construction of this bijection using an abacus. For a partition $\lambda$, define the infinite set ${\beta }^{\lambda }=\{{\lambda }_i-i\mid i\ge 1\}$ where ${\lambda }_i=0$ for every $i>k$. Given the set $\beta$, we can recover $\lambda$ as follows: shift all entries of $\beta$ so that Template:Math 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 $\lambda$.

Consider an abacus with Template:Mvar infinitely long vertical runners numbered Template:Math, Template:Math, up to Template:Math. Label the position on runner $r$ at height $s$ by $st+r$, so values increase left-to-right then bottom-to-top.

Given a partition $\lambda$, place beads on the abacus at each position in ${\beta }^{\lambda }$. If ${\beta }_{(r)}$ are the heights of the beads on runner $r$, then ${\lambda }_{(r)}$ is the unique partition with ${\beta }^{{\lambda }_{(r)}}={\beta }_{(r)}$. Next, suppose ${\beta }^{\prime }$ are the positions of the beads when the beads in ${\beta }^{\lambda }$ naturally fall under gravity. Then ${\lambda }^{(t)}$ is the unique partition satisfying ${\beta }^{{\lambda }^{(t)}}={\beta }^{\prime }$.

Example

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

We draw our Template:Math-abacus by circling the beads in ${\beta }^{\lambda }$.

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

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

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

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

Other identities

Ramanujan's modular equations can be used to prove identities for $c_t(n)$, such as $c_3(4n+1)=c_3(n)$ and $c_5(4n+3)=c_5(2n+1)+2c_5(n)$.

Partitions which are simultaneously Template:Mvar-core for multiple values of Template:Mvar are well-studied. For example, if Template:Mvar and Template:Mvar are coprime positive integers, then the number of partitions which are simultaneously Template:Mvar-core and Template:Mvar-core is equal to $$\frac{1}{s+t}(\genfrac{}{}{0ex}{}{s+t}{s}),$$ which is a rational Catalan number.

The number of Template:Mvar-core partitions with at most Template:Mvar rows is equal to the number of partitions with at most Template:Mvar rows and at most Template:Math columns. A bijection between these sets is given by $\lambda \mapsto \partial$, where ${\partial }_i$ is the number of cells in row $i$ of $\lambda$ whose hook length is less than Template:Mvar.

See also

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