Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k

TL;DR

This paper generalizes previous results on partition parts by analyzing the count of parts divisible by k in partitions with limited multiplicity, providing generating functions and combinatorial proofs.

Contribution

It extends known results to a broader class of partitions with multiplicity constraints, introducing new generating functions and combinatorial proofs.

Findings

Derived generating functions for the new partition counts

Provided combinatorial proofs using extended Glaisher's bijection

Verified recurrence relations for the new sequences

Abstract

Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the partitions of n for which the multiplicity of each part is strictly less than k, ak(n). Moreover, a combinatorial proof is provided using an extension of Glaisher's bijection. Finally, we give the generating functions for this new family of integer sequences and use it to verify generalized pentagonal, triangular, and square power recurrence relations.