DEFINITION. Let $t \geq 1$ be a positive integer. Any partition of a positive integer $n$ whose Ferrers graph has no hook numbers divisible by $t$ is known as a $t$−core partition, or $t$-core of $n$.

For $t \geq 1$ and $n \geq 0$, the number of $t$−core partitions of $n$ is denoted by $c_t(n)$ ($c_t(0)$ is defined to be 1 for all $t\geq 1$). The ordinary generating function for the sequence $\{c_t(n)\}_{n=0}^{\infty}$ is given by \begin{equation}\label{tcoreq1} \sum_{n=0}^{\infty}c_t(n)q^n=\prod_{n=1}^{\infty}\frac{(1-q^{tn})^t}{1-q^n}. \end{equation} An obvious question is: Given a positive integer $t$, is $c_t(n) > 0$ for all $n \geq 0$?

This can be shown to be false for $t=2,3$. Indeed, from the theory of $q$-series (in particular, the Jacobi triple product identity) it is well-known that \begin{equation}\label{tcoreq2} \sum_{n=0}^{\infty}c_2(n)q^n=\prod_{n=1}^{\infty}\frac{(1-q^{2n})^2}{1-q^n} = \sum_{n=0}^{\infty}q^{n(n+1)/2}. \end{equation} Thus a reasonable conjecture might be that $c_t(n) > 0$ for all $n \geq 0$ and all $t>3$.

Since every $t$-core is a $tk$-core (if the Ferrers graph of a partition contains no hook numbers that are multiples of $t$, then it will have no hook numbers that are multiples of $tk$), and thus $c_t(n)\leq c_{tk}(n)$, it would sufficient to prove this for $t=4, 6, 9$ and any prime greater than 3.

In [GKS90], Garvan, Kim and Stanton give explicit formulae for $c_5(n)$ and $c_7(n)$, which show that $c_5(n)>0$ and $c_7(n)>0$ for all $n\geq 0$. Their formula for $c_5(n)$ is contained in the following theorem (notation slightly changed here for consistency).

THEOREM (Garvan, Kim and Stanton 1990). Let \[ n+1 = 5^c p_1^{a_1}\dots p_s^{a_s} q_1^{b_1}\dots q_t^{b_t} \] be the prime factorization of $n+1$ into primes $p_i \equiv 1,4 \mod 5$, and $q_j \equiv 2,3 \mod 5$. Then \begin{equation}\label{tcoreq3} c_5(n) = 5^c \prod_{i=1}^s \frac{p_i^{a_i +1}− 1} {p_i − 1} \prod_{j=1}^t \frac{q_j^{b_j +1} + (−1)^{b_j}} {q_j +1}. \end{equation} By using the theory of modular forms, Ono proved the following theorem (a result also proved by Klyachko [K84]).

THEOREM. If $t$ is a positive integer with at least one prime factor $p \geq 5$, then \begin{equation}\label{tcoreq4} c_t (n) = 0 \text{ for at most finitely many } n \geq 0. \end{equation} By additional calculations, Ono showed that in fact $c_{11}(n)>0$ for all $n\geq 0$.

In the same paper [O94], Ono also proved positivity results for $t$ a multiple of 4 and for $t=9$ (which implies the positivity of $c_t(n)$ for $t$ a multiple of 9).

THEOREM. If $t$ is a multiple of 4, then $c_t (n) > 0$ for all $n \geq 0$.

THEOREM. If $c_9(n)$ is the number of 9-core partitions of $n$, then $c_9(n) > 0$ for all $n \geq 0$.

In [O95], Ono showed that $c_t(n)>0$ if $3|t$.

The authors in [GKS90] also prove a number of equalities and congruence formulae for $c_5(n)$ and $c_7(n)$, namely that for all integers $n\geq 0$ and all integers $\alpha \geq 1$, there holds \begin{align}\label{tcoreq5} c_5 (5^{\alpha} m − 1) &= 5^{\alpha} c_5 (m − 1) \equiv 0 (\mod 5^{\alpha}),\\ c_7 (7^{\alpha} m − 2) &\equiv 0 (\mod 7^{\alpha}),\notag\\ c_7 (49n + 19) &= 49c_7 (7n + 1),\notag\\ c_7 (49n + 33) &= 49c_7 (7n + 3),\notag\\ c_7 (49n + 40) &= 49c_7 (7n + 4).\notag \end{align}

It is also possible to consider restricted types of partitions which are also $t$-cores. In [GKS90] the authors considered self-conjugate partitions which are $t$-cores, and partitions into distinct parts which are $t$-cores, deriving generating functions for these, and also results similar to those at \eqref{tcoreq5}.

Other generalizations and extensions of the concept of a $t$-core partition have also been considered.

[GKS90] F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), 1–17.

[K82] A. A. Klyachko, Modular forms and representations of symmetric groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 74–85 (in Russian).

[O94] K. Ono, On the positivity of the number of t-core partitions. Acta Arith. 66 (1994), no. 3, 221–228.

[O95] K. Ono, A note on the number of t-core partitions. Rocky Mountain J. Math. 25 (1995), no. 3, 1165–1169.

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