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# Wydawnictwa / Czasopisma IMPAN / Acta Arithmetica / Wszystkie zeszyty

## On the positivity of the number of t-core partitions

### Tom 66 / 1994

Acta Arithmetica 66 (1994), 221-228 DOI: 10.4064/aa-66-3-221-228

#### Streszczenie

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. \de{1.} Let $t ≥ 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partition} of $n.$ \vskip 4pt plus 2pt The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan's congruences for the ordinary partition function [3,$\,$4,$\,$6]. If $t≥ 1$ and $n ≥ 0$, then we define $c_t(n)$ to be the number of partitions of n that are t-core partitions. The arithmetic of $c_t(n)$ is studied in [3,$\,$4]. The power series generating function for $c_t(n)$ is given by the infinite product: ∑_{n=0}^{∞} c_t(n)q^n= \prod_{n=1}^{∞

• Ken Ono

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