Arithmetical properties of the number of t-core partitions

Let I >={lambda (1)a parts per thousand yena <...a <...a <...a parts per thousand yen lambda (s) a parts per thousand yen1} be a partition of an integer n. Then the Ferrers-Young diagram of I > is an array of nodes with lambda (i) nodes in the ith row. Let lambda (j) ' denote the number of nodes in column j in the Ferrers-Young diagram of I >. The hook number of the (i,j) node in the Ferrers-Young diagram of I > is denoted by H(i,j):=lambda (i) +lambda (j) '-i-j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 (t) -core partitions of n are obtained. A simple convolution identity for t-cores is also given.