Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part

Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(π)$, appears $k$ times and every overlined part is bigger than $s(π)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part appears $k$ times, every overlined part is bigger than $s(π)$ and all parts other than $s(π)$ are incongruent modulo $2$ with $s(π)$. Also, let $b_e(k,n)$ (resp., $b_o(k,n)$) denote the number of overpartitions of $n$ counted by $\overline{\mathrm{spt}}k_o(n)$ where the number of parts greater than $s(π)$ is even (resp., odd), and let $$\overline{\mathrm{spt}}k_o’(n)=b_e(k,n)-b_o(k,n).$$ Recently, Malik and Sarma (arXiv:2601.15601v1) expressed the generating functions of these partition functions in terms of linear combinations of $q$-series with polynomials in $q$ as coefficients. As corollaries, they derived some partition identities involving the functions for $k=1$ and sought for combinatorial proofs of their results. In this paper, we present some desired proofs.

💡 Research Summary

This paper investigates a family of overpartitions in which the smallest non‑overlined part, denoted (s(\pi)), occurs exactly (k) times and every overlined part is larger than (s(\pi)). The authors focus on the case (k=1) and provide purely combinatorial (bijection‑based) proofs of three identities recently obtained analytically by Malik and Sarma (arXiv:2601.15601v1). In addition, they present a refined theorem (Theorem 1.4) that simultaneously yields the two identities for (k=1) and explains the third identity as a corollary.

Main objects.

• (\overline{\mathrm{spt}}_k(n)): number of overpartitions of (n) with smallest non‑overlined part appearing exactly (k) times and all overlined parts larger than that smallest part.
• (\overline{\mathrm{spt}}_{k,o}(n)): same as above, with the extra condition that every part different from (s(\pi)) has opposite parity modulo 2 from (s(\pi)).
• (p_e(n)): number of overpartitions of (n) into even parts only.
• (p_{ex}(n)): number of overpartitions of (n) with no non‑overlined 1.
• (p_{oex}(n)): number of overpartitions of (n) into odd parts only and with no non‑overlined 1.
• (b_e(k,n)) and (b_o(k,n)): among the overpartitions counted by (\overline{\mathrm{spt}}_{k,o}(n)), the subsets where the number of parts larger than (s(\pi)) is even (resp. odd).
• (\overline{\mathrm{spt}}’_{k,o}(n)=b_e(k,n)-b_o(k,n)).

Malik and Sarma derived, via generating‑function manipulations, the following identities for (k=1):

1. (\overline{\mathrm{spt}}_1(n)+\overline{\mathrm{spt}}1(n-1)=p{ex}(n)) for (n>1).
2. (\overline{\mathrm{spt}}{1,o}(n)+\overline{\mathrm{spt}}{1,o}(n-2)=2p_e(n-1)+p_{oex}(n-1)) for (n>2).
3. (\overline{\mathrm{spt}}’{1,o}(n)+\overline{\mathrm{spt}}’{1,o}(n-2)=-p’_{oex}(n-1)) for (n>2).

These results were proved analytically; the authors of the present paper set out to give bijective proofs, thereby providing a deeper combinatorial understanding.

Bijection strategy.
The authors construct explicit maps between the relevant families of overpartitions. The maps are defined case‑by‑case according to the value of the smallest part (s(\pi)) and the parity of (s(\pi)). The key observations are:

• Decreasing or increasing the smallest part by one (while preserving the overline status) often moves a partition from one family to another without creating overlaps.
• The parity of (s(\pi)) determines whether the image lands in a set of even‑part overpartitions ((p_e)) or in the set of odd‑part overpartitions without non‑overlined 1 ((p_{oex})).
• For the signed count (\overline{\mathrm{spt}}’_{1,o}(n)), the authors introduce the notions of “SAME nature” and “OPPOSITE nature” (borrowed from their earlier work). Partitions of opposite nature contribute with opposite signs, which explains the negative sign appearing in identity 3.

Proof of Theorem 1.1.
Four maps (f_1,\dots,f_4) are defined from (\mathrm{Spt}1(n)) and (\mathrm{Spt}1(n-1)) to (\mathrm{P{ex}}(n)). Roughly: if (s(\pi)>1) we keep the partition unchanged; if (s(\pi)=1) we either overline the 1 or replace it by a 2, depending on whether the next larger part is adjacent. The four images are disjoint and together exhaust (\mathrm{P{ex}}(n)), establishing the identity.

Proof of Theorem 1.2.
Partitions in (\mathrm{Spt}{1,o}(n)) with (s(\pi)=1) are reduced by deleting the 1, yielding a bijection onto (\mathrm{P_e}(n-1)). For the remaining partitions, the parity of (s(\pi)) dictates whether we replace (s(\pi)) by (s(\pi)-1) (if even) or (s(\pi)+1) (if odd), sending the result into (\mathrm{P{oex}}(n-1)) or (\mathrm{P_e}(n-1)) respectively. This produces the required count (2p_e(n-1)+p_{oex}(n-1)).

Proof of Theorem 1.3.
Using the SAME/OPPOSITE classification, the authors construct a sign‑reversing involution on (\mathrm{Spt}{1,o}(n)\cup\mathrm{Spt}{1,o}(n-2)). The involution either increases or decreases the second smallest part (s_2(\pi)) by one, flipping the nature of the partition. The fixed points correspond precisely to overpartitions counted by (p’_{oex}(n-1)), and the sign reversal yields the negative sign in the identity.

Theorem 1.4 (Refined version).
The authors prove two equalities: \