Draft:Core Partition

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 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 of each cell 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

${\displaystyle \sum _{n=0}^{\infty }c_{t}(n)q^{n}={\frac {\phi (q^{t})^{t}}{\phi (q)}}}$

where ${\displaystyle \phi (q)=(1-q)(1-q^{2})(1-q^{3})\cdots }$ 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 ${\displaystyle (\lambda ^{(t)},\Lambda _{(t)})}$, where ${\displaystyle \lambda ^{(t)}}$ is a t-core partition and ${\displaystyle \Lambda _{(t)}=(\lambda _{(1)},\dots ,\lambda _{(t-1)})}$ is a sequence of partitions such that

${\displaystyle |\lambda |=|\lambda ^{(t)}|+t(|\lambda _{(0)}|+\cdots +|\lambda _{(t-1)}|).}$

We call ${\displaystyle \lambda ^{(t)}}$ the t-core of ${\displaystyle \lambda }$ and ${\displaystyle \Lambda ^{(t)}}$ the t-quotient of ${\displaystyle \lambda }$.

Example

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 hooklengths ${\displaystyle \{5,2,1\}}$ which means ${\displaystyle \lambda ^{(4)}=(5,1,1)}$, 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