<i>n</i>-Color Partitions into Distinct Parts as Sums over Partitions

Abstract

The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For r∈{0,1}, let QLr(n) be the number of n-color partitions of n into distinct parts which have a number of parts congruent to r modulo 2. In this paper, we consider specializations of complete and elementary symmetric functions in order to establish two kinds of formulas for QL0(n)±QL1(n) as sums over partitions of n in terms of binomial coefficients. The first kind of formulas only involve partitions in which the parts of size n appear at most n times, while the second kind of formulas involve unrestricted partitions. Similar results are obtained for the first differences of QL0(n)±QL1(n) and the partial sums of QL0(n)±QL1(n).

Keywords

• partitions
• symmetric functions
• binomial coefficients