Number of partitions of n with a given parity of the smallest part

TL;DR

This paper provides a combinatorial proof of a partition equality and introduces an efficient method to count partitions of n based on the parity of their smallest part, connecting to rank-based partition statistics.

Contribution

It offers a natural combinatorial proof of a weighted partition equality and derives a practical formula for counting partitions with a specified smallest part parity.

Findings

Established a combinatorial proof of a partition equality

Derived a formula linking smallest part parity to rank-based partition counts

Presented an efficient method for counting partitions with parity constraints

Abstract

We obtain a combinatorial proof of a surprising weighted partition equality of Berkovich and Uncu. Our proof naturally leads to a formula for the number of partitions with a given parity of the smallest part, in terms of S(i), the number of partitions of i into distinct parts with even rank minus the number with odd rank, for which there is an almost closed formula by Andrews, Dyson and Hickerson. This method of calculating the number of partitions of n with a given parity of the smallest part is practical and efficient.