In combinatorial mathematics, a -core partition is a partition which has no hooks of length . 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 is a weakly decreasing list of positive integers, which we associate with its Young diagram by drawing square cells in row of an array. The size of a partition, denoted , is the sum of all , or the total number of cells. The conjugate partition is the partition whose values are the length of each column in .
The cells in the diagram are labelled by for and . The hook length of cell is given by
which is equal to the number of cells in the rotated L-shaped hook with vertex at which extends to the right and downwards.
For a positive integer , a partition is -core if it has no cells with a hook length of .
Properties
If a partition is -core, then it is also ()-core for every positive integer .
In any row/column of a -core partition which contains a cell of hook length > , there is a cell in the same row/column with hook length – .
The only -core partition is the empty partition with no cells. The -core partitions are the staircase partitions . If is the perimeter of the Young diagram of a partition, then this partition is -core for every .
+ Number of -core partitions of size . 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 at least , there exists a -core partition of size for every positive integer . This result was known as the -core conjecture before finally being proven by Andrew Granville and Ken Ono in 1996.
If is the number of -core partitions of size , we have the generating function
where is the Euler function. Note that is the generating function for all partitions.
Abacus
The generating function for -core partitions implies that there is a bijection between partitions and pairs , where is a -core partition and is a sequence of partitions such that
We call the -core of and the -quotient of .
Construction
We give a construction of this bijection using an abacus. For a partition , define the infinite set where for every . Given the set , we can recover as follows: shift all entries of so that 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 .
Consider an abacus with infinitely long vertical runners numbered , , up to . Label the position on runner at height by , so values increase left-to-right then bottom-to-top.
Given a partition , place beads on the abacus at each position in . If are the heights of the beads on runner , then is the unique partition with . Next, suppose are the positions of the beads when the beads in naturally fall under gravity. Then is the unique partition satisfying .
Example
Suppose and . Then
We draw our -abacus by circling the beads in .
Looking at runner (the first column), the shaded beads have heights . Hence, the first-column hook lengths of are , and so
In runner , we have and so is the empty partition . We have so , and finally . These partitions make up the -quotient .
Now we calculate the -core of . Letting the beads of fall under gravity gives the abacus:
Therefore,
The smallest missing value is , so shifting the values by gives the first-column hook lengths which means , which is indeed a -core partition.
Other identities
Ramanujan's modular equations can be used to prove identities for , such as and .
Partitions which are simultaneously -core for multiple values of are well-studied. For example, if and are coprime positive integers, then the number of partitions which are simultaneously -core and -core is equal to
which is a rational Catalan number.
The number of -core partitions with at most rows is equal to the number of partitions with at most rows and at most columns. A bijection between these sets is given by , where is the number of cells in row of whose hook length is less than .
See also Hook length formula Representation theory of the symmetric group Crank of a partition
References