Core partition

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 and for representation theory of the symmetric group, especially modular representation theory.

Definition

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

The cells in the diagram are labelled by $(i, j)$ for $1 \leq i \leq k$ and $1 \leq j \leq λ_{i}$ . The hook length $h_{λ} (i, j)$ of cell $(i, j)$ is given by $h_{λ} (i, j) = λ_{i} + {λ_{j}}^{'} - i - j + 1,$ 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 $t$ , a partition is $t$ -core if it has no cells with a hook length of $t$ .

Properties

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 $h$ – $t$ .

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

Number of $t$ -core partitions of size $n$ .
$n$ $t$	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 $t$ 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.

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

Abacus

The generating function for $t$ -core partitions implies that there is a bijection between partitions $λ$ and pairs $(λ^{(t)}, Λ_{(t)})$ , where $λ^{(t)}$ is a $t$ -core partition and $Λ_{(t)} = (λ_{(1)}, \dots, λ_{(t - 1)})$ is a sequence of partitions such that $| λ | = | λ^{(t)} | + t (| λ_{(0)} | + \dots + | λ_{(t - 1)} |) .$ We call $λ^{(t)}$ the $t$ -core of $λ$ and $Λ^{(t)}$ the $t$ -quotient of $λ$ .

Construction

We give a construction of this bijection using an abacus. For a partition $λ$ , define the infinite set $β^{λ} = {λ_{i} - i ∣ i \geq 1}$ where $λ_{i} = 0$ for every $i > k$ . Given the set $β$ , we can recover $λ$ as follows: shift all entries of $β$ 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 $λ$ .

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

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

Example

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

We draw our $4$ -abacus by circling the beads in $β^{λ}$ .

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

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

Now we calculate the $4$ -core of $λ$ . Letting the beads of $β^{λ}$ fall under gravity gives the abacus:

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

Other identities

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

Partitions which are simultaneously $t$ -core for multiple values of $t$ are well-studied. 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 $\frac{1}{s + t} (\binom{s + t}{s}),$ which is a rational Catalan number.

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 $λ \mapsto \partial$ , where $\partial_{i}$ is the number of cells in row $i$ of $λ$ whose hook length is less than $t$ .

References

↑ Garvan, Frank; Kim, Dongsu; Stanton, Dennis (1990). "Cranks and t-cores". Invent. Math. 101 (1): 1–17.
↑ James, Gordon; Kerber, Albert (1981). The representation theory of the symmetric group. Reading, Mass.: Addison-Wesley Publishing Co.
↑ Olsson, Jørn B.; Stanton, Dennis (2007). "Block inclusions and cores of partitions". Aequationes Math. 74 (1–2): 90–110.
1 2 James, Gordon (1978). "Some combinatorial results involving Young diagrams". Math Proc. Cambridge Philos. Soc. 83 (1): 1–10.
↑ Granville, Andrew; Ono, Ken (1996). "Defect zero p-blocks for finite simple groups". Trans. Amer. Math. Soc. 348 (1): 331–347.
↑ Baruah, Nayandeep; Berndt, Bruce (2007). "Partition identities and Ramanujan's modular equations". J. Combin. Theory Ser. A. 114 (6): 1024–1045.
↑ Cho, Hyunsoo; Kim, Byungchan; Nam, Hayan; Sohn, Jaebum (2021). "A survey on t-core partitions". Hardy-Ramanujan Journal. 44: 81–101.
↑ Anderson, Jaclyn (2002). "Partitions which are simultaneously t1- and t2-core". Discrete Math. 248 (1–3): 237–243.
↑ Lapointe, Luc; Morse, Jennifer (2005). "Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions". Journal of Comb. Thy., Series A. 112 (1): 44–81.

[1] Garvan, Frank; Kim, Dongsu; Stanton, Dennis (1990). "Cranks and t-cores". Invent. Math. 101 (1): 1–17.

[2] James, Gordon; Kerber, Albert (1981). The representation theory of the symmetric group. Reading, Mass.: Addison-Wesley Publishing Co.

[3] Olsson, Jørn B.; Stanton, Dennis (2007). "Block inclusions and cores of partitions". Aequationes Math. 74 (1–2): 90–110.

[james-4] 1 2 James, Gordon (1978). "Some combinatorial results involving Young diagrams". Math Proc. Cambridge Philos. Soc. 83 (1): 1–10.

[5] Granville, Andrew; Ono, Ken (1996). "Defect zero p-blocks for finite simple groups". Trans. Amer. Math. Soc. 348 (1): 331–347.

[6] Baruah, Nayandeep; Berndt, Bruce (2007). "Partition identities and Ramanujan's modular equations". J. Combin. Theory Ser. A. 114 (6): 1024–1045.

[7] Cho, Hyunsoo; Kim, Byungchan; Nam, Hayan; Sohn, Jaebum (2021). "A survey on t-core partitions". Hardy-Ramanujan Journal. 44: 81–101.

[8] Anderson, Jaclyn (2002). "Partitions which are simultaneously t1- and t2-core". Discrete Math. 248 (1–3): 237–243.

[9] Lapointe, Luc; Morse, Jennifer (2005). "Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions". Journal of Comb. Thy., Series A. 112 (1): 44–81.