On the positivity of the number of t-core partitions

A partition of a positive integer n is a nonincreasing sequence of positive integers with sum ${\displaystyle n.}$ Here we define a special class of partitions. \de{1.} Let ${\displaystyle t\ge 1}$ 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 ${\displaystyle n.}$ \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,${\displaystyle ,}$4,${\displaystyle ,}$6]. If ${\displaystyle t\ge 1}$ and ${\displaystyle n\ge 0}$, then we define ${\displaystyle c_t\left(n\right)}$ to be the number of partitions of n that are t-core partitions. The arithmetic of ${\displaystyle c_t\left(n\right)}$ is studied in [3,${\displaystyle ,}$4]. The power series generating function for ${\displaystyle c_t\left(n\right)}$ is given by the infinite product: ∑_{n=0}^{∞} c_t(n)q^n= \prod_{n=1}^{∞