Combinatorial proofs for partitions with repeated smallest part

TL;DR

This paper provides combinatorial (bijective) proofs for identities about integer partitions in which the smallest part is repeated $k$ times, with either the remaining parts distinct or incongruent modulo $2$ to the smallest part. The authors rewrite the analytic results of Andrews and El Bachraoui as combinatorial statements and construct explicit bijections to prove key recurrences, notably $spt_k_d(n)+spt(k-1)_d(n)=spt(k-1)_d(n-k+1)+p_d(n-k+1)$ and related relations for the distinct-even/odd variants. They establish the corresponding generating-function forms, show how the results imply the claimed corollaries, and connect the proofs for the $d$- and $do$-families via a unified bijective framework. The work strengthens the combinatorial understanding of $Spt$ statistics in partitions with constrained repetition and provides the first purely combinatorial proofs of these identities and corollaries. The methods offer a template for deriving further partition identities through explicit bijections across partition classes.

Abstract

Recently, Andrews and El Bachraoui considered the number of integer partitions whose smallest part is repeated exactly $k$ times and the remaining parts are not repeated. They presented several interesting results and posed questions regarding combinatorial proofs for these identities. In this paper, we establish bijections to provide combinatorial proofs for these results.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 2.1
• proof
• Lemma 3.1
• proof