Proofs of Two Formulas of Vladeta Jovovic

TL;DR

This paper analyzes partitions whose smallest part occurs at least twice, focusing on two formulas attributed to Vladeta Jovovic. It delivers an analytic proof of $a(n)=2p(n)-p(n+1)$ and two bijective proofs: a direct bijection proving $a(n)=2p(n)-p(n+1)$ and a bijection proving $a(n)=p(2n,n)$, linking the partition statistic to fixed-difference partitions. The methods combine generating function techniques, $q$-hypergeometric identities, and explicit combinatorial bijections between well-defined partition classes. The results establish a precise equivalence between these two natural partition statistics and suggest directions for generalizations via $a_m(n)$ and related generating functions.

Abstract

In this paper, we first provide an analytic and a bijective proof of a formula stated by Vladeta Jovovic in the OEIS sequence A117989. We also provide a bijective proof of another interesting result stated by him on the same page concerning integer partitions with fixed differences between the largest and smallest parts.