Separable overpartition classes and excludant sizes of an overpartition

An overpartition is a partition such that the first occurrence (equivalently, the last occurrence) of a number may be overlined. In this article, we investigate three contents of overpartitions. We first consider the $r$-chain minimal and maximal excludant sizes of an overpartition. Then, we study the second minimal excludant and mex sequence of an overpartition. Finally, we introduce $L_k$-overpartitions and $F_k$-overpartitions, which are separable overpartition classes.

💡 Research Summary

The paper investigates three distinct aspects of overpartitions, extending several classical partition statistics to this richer combinatorial object and introducing new separable classes.

First, the authors generalize the minimal excludant (mex) to an r‑chain minimal excludant size, denoted mes(π;r). This is the smallest positive integer t such that none of the consecutive integers t, t+1, …, t+r−1 appear as parts of the overpartition π. Dually, they define the r‑chain maximal excludant size, maes(π;r), as the largest integer t smaller than the largest part of π for which the block t, t−1, …, t−r+1 is completely absent; if no such t exists, maes(π;r)=0. For each n they consider the total sums σ_r^mes(n)=∑{π∈P(n)}mes(π;r) and σ_r^maes(n)=∑{π∈P(n)}maes(π;r). By employing the auxiliary series ω(t)=1+∑{i=1}^r q^{it} and standard q‑product notation, they derive explicit generating functions (equations (1.1) and (1.2)). These formulas involve the infinite products (−q;q)∞ and (q;q)_∞ together with finite sums of q‑powers, providing a compact analytic description of the distribution of these chain‑excludant statistics.

Second, the paper introduces the second minimal excludant mex₂(π) for overpartitions, defined as the smallest integer larger than mex(π) that is missing from the part multiset. The authors also define the mex sequence of an overpartition as the longest run of consecutive missing integers beginning at mex(π). They study two aggregate quantities: σ₂^mex(n)=∑{π∈P(n)}mex₂(π) and Δ_t(n)=|{π∈P(n): mex₂(π)−mex(π)=t}|. The generating function for σ₂^mex(n) is given in equation (1.12) and combines the basic product (−q;q)∞(q;q)∞ with a weighted sum over s involving factorial-like q‑coefficients. For Δ_t(n) they obtain equation (1.13), which features the factor (q^{t−1}−q^{t}) multiplied by the same infinite products and a double‑sum over m, reflecting the combinatorial constraint on the gap between mex and mex₂. Moreover, they define p{mex}^r(n) as the number of overpartitions whose mex sequence has length at least r, prove that p_{mex}^r(n) equals the number p_r(n) of overpartitions satisfying a certain bounded‑multiplicity condition, and give a generating function (Theorem 1.16) expressed via q‑binomial coefficients and the product (−2q^m+1;q)_∞.

Third, the authors turn to structural classifications. They recall the notion of a separable integer partition class (a set of partitions that can be uniquely expressed as a sum of a “basis” partition and a non‑increasing sequence of non‑negative integers). Extending this to overpartitions, they define two families parameterized by a modulus k:

• L_k‑overpartitions: overpartitions in which any overlined part occurs only at positions whose distance from the end of the partition is a multiple of k (i.e., if part π_i is overlined then ℓ−i ≡ 0 (mod k)).
• F_k‑overpartitions: overpartitions in which any overlined part occurs only at positions whose distance from the beginning satisfies ℓ−i ≡ −1 (mod k).

Both families satisfy the separability condition: each overpartition can be uniquely written as (λ₁+μ₁)+(λ₂+μ₂)+… where (λ₁,…,λ_m) belongs to a finite basis B (depending on k) and (μ₁,…,μ_m) is a non‑increasing sequence of non‑negative integers. Using this decomposition, the authors derive closed‑form generating functions for the total weight (by size) and the number of overlined parts (ℓ_o) in each class. The generating functions are given in equations (1.5) and (1.6) and involve q‑binomial coefficients (Gaussian polynomials) together with the basic infinite products (−q;q)∞ and (q;q)∞.

The paper concludes with brief outlines of the proofs. Section 2 introduces the conjugate of an overpartition and the (r+1)-rep‑eating size, establishing bijections that relate mes and maes to these auxiliary statistics. Sections 3 and 4 apply q‑series manipulations, the theory of basic hypergeometric series, and the separable class framework to obtain the generating functions stated earlier.

Overall, the work enriches the theory of overpartitions by (i) extending mex‑type statistics to r‑chains and second‑order gaps, (ii) providing precise q‑generating functions for the associated distributions, and (iii) constructing new separable families L_k and F_k that generalize classical parity‑separated partition theorems to the overpartition setting. These results open avenues for further exploration of modular identities, Rogers–Ramanujan‑type phenomena, and potential applications in statistical mechanics models where overpartition structures naturally arise.