A direct bijective proof of the odd/even minimal-excludant equidistribution with sqrank and rerank, and a succinct partition statistic for $p_{2,2}(n)$

This task concerns the open directions in Taichiro Takagi's study of the odd/even minimal excludant (mex) in integer partitions (Annals of Combinatorics, 2026). Takagi introduced two partition statistics, sqrank and rerank, defined through Durfee-square/rectangle decomposition, iterated rim-hook stripping, and a maximum over Frobenius coordinates, and proved analytically - via a polynomial bosonic form of one-dimensional configuration sums originating in the box-ball system - that the partitions of $n$ with odd (resp. even) minimal excludant $2r+1$ (resp. $2r+2$) are equinumerous with the partitions of $n$ on which sqrank (resp. rerank) takes the value $r$, thereby answering a question of Andrews and Newman on the mex-counting function $p_{2,2}(n)$. The paper itself expresses two hopes that remain unfulfilled: a direct, explicit bijective proof of this equidistribution, and a partition statistic of rank/crank-like brevity that reproduces $p_{2,2}(n)$. A 2026 follow-up by Miyazawa and Takagi constructs energy-preserving bijections between the statistic classes and highest-weight paths of level-one $U_q(\widehat{\mathfrak{sl}}_2)$ crystals, but its connection to the mex classes still passes through the analytic branching-function identity, so both asks remain open.

Background

An integer partition of $n$ is a weakly decreasing sequence of positive integers summing to $n$, drawn as a Ferrers diagram of unit squares whose row lengths are the parts. Two standard diagrammatic notions are used below. The Durfee square of a partition is the largest square sitting in the upper-left corner of its Ferrers diagram; the Durfee rectangle (as used here) is the largest rectangle containing that corner whose width exceeds its height by exactly one. A rim hook (border strip) is a connected skew diagram containing no $2\times 2$ block of cells. The Frobenius representation of a partition $\lambda$ with Durfee square of side $d$ is $F(\lambda)=(x_1,\dots,x_d\mid y_1,\dots,y_d)$, where $x_i$ (resp. $y_i$) counts cells of row $i$ to the right of (resp. column $i$ below) the diagonal cell $(i,i)$.

Andrews and Newman (2020) extended the minimal excludant (mex, the least positive integer missing from a set) to partitions: $\mathrm{mex}_{A,a}(\lambda)$ is the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\lambda$. They studied $p_{A,a}(n)$, the number of partitions of $n$ whose $\mathrm{mex}_{A,a}$ lies in the residue class $a$ modulo $2A$, and, for $A=2$, the refinements $p^{(r)}_{2,1}(n)$ and $p^{(r)}_{2,2}(n)$ counting partitions with $\mathrm{mex}_{2,1}(\lambda)=2r+1$ and $\mathrm{mex}_{2,2}(\lambda)=2r+2$, which were introduced in Takagi's study. In particular $\mathrm{mex}_{2,1}$ is the smallest missing odd positive integer and $\mathrm{mex}_{2,2}$ the smallest missing even positive integer (the smallest positive integer congruent to $2$ modulo $2$), so $p_{2,2}(n)$ counts partitions of $n$ whose even minimal excludant is congruent to $2$ modulo $4$ (that is, is not a multiple of four). They proved that $p_{1,1}(n)$ equals the number of partitions of $n$ with nonnegative crank, that $p_{3,3}(n)$ equals the number with rank at least $-1$ (rank and crank being the classical statistics of partition theory, defined in a few lines from the parts: rank is the largest part minus the number of parts, and crank is built from the multiplicity of the part 1 and the number of parts), and that $p_{2,1}(n)$ equals $p_e(n)$, the number of partitions of $n$ into an even number of parts. They then asked whether a comparable theorem exists for $p_{2,2}(n)$ (their Question II), and, studying the generating function

$\sum_{n\geq 0}p_{2,2}(n)q^{n}=\sum_{n\geq 0}\frac{q^{n^{2}+n}B_{n}(q)}{(q;q)_{2n+1}},\qquad B_{n}(q)=\sum_{r=0}^{n}(-1)^{r}{2n+1\brack n-r},$

where $(q;q)_m=\prod_{k=1}^{m}(1-q^k)$ and ${L\brack s}$ is the Gaussian polynomial (the $q$-binomial coefficient generating partitions in an $s\times(L-s)$ box), they conjectured that $B_n(q)$ has nonnegative coefficients (proved by Chern, 2023), is unimodal, and enumerates some subset of the partitions into at most $n$ parts each at most $n+1$.

Takagi (arXiv:2412.19503, published in Annals of Combinatorics, 2026) answered Question II. He introduced two statistics defined by the following procedure on the Ferrers diagram of $\lambda$ (for $a=0,1$): let $D_a(\lambda)$ be the Durfee square ($a=0$) or Durfee rectangle ($a=1$), of height $n_a$, and let $A_a(\lambda)$ be the diagram of the first $n_a$ rows with $D_a$ removed; repeatedly strip from the rim of $A_a$ the longest rightmost rim hook of arm length $n_a+a$ until fewer than $n_a+a+1$ columns remain, leaving a residual diagram $R_a$; then, with $F(R_a)=(x_1,\dots,x_d\mid y_1,\dots,y_d)$, $y_0=n_a$ and $x_{d+1}=-1$, set $f^{(a)}(\lambda)=\max_{0\leq i\leq d}(y_i-x_{i+1})-1$. The statistics are $\mathrm{sqrank}(\lambda)=f^{(0)}(\lambda)$ and $\mathrm{rerank}(\lambda)=f^{(1)}(\lambda)$ (with the conventions that $D_a$, $A_a$ are empty and $n_a=0$ when $\lambda=\emptyset$, or, for $a=1$, when $\lambda$ has only ones as parts, and that $f^{(a)}(\lambda)=n_a$ when the residual diagram $R_a$ is empty). His Theorem 1 states that $p^{(r)}_{2,1}(n)$ equals the number of partitions of $n$ with sqrank $r$, and $p^{(r)}_{2,2}(n)$ the number with rerank $r$; as a corollary, $p_{2,2}(n)$ equals the number of partitions of $n$ with even rerank. The proof is analytic: the key polynomial $X^{+}_{L,s}(q)={L\brack s}-{L\brack s-1}$, a difference of Gaussian polynomials (a so-called bosonic form, i.e. a manifestly subtractive expression that nevertheless has nonnegative coefficients), is rewritten as a one-dimensional configuration sum $\sum_{\eta}q^{E(\eta)}$ over bit sequences with an energy $E$, a type of formula originating in exactly solved lattice models and conformal-field-theory characters; Takagi's derivation draws on his earlier work on the box-ball system, an integrable cellular automaton whose soliton content is encoded by such bit sequences. Along the way he identifies the subset enumerated by $B_n(q)$: writing $\mathcal{E}_1(\lambda;n)=\max_{0\leq i\leq d}(y_i-x_{i+1})-1$ on Frobenius coordinates with $y_0=n$, $x_{d+1}=-1$, the polynomial $B_n(q)$ enumerates the partitions in the $n\times(n+1)$ box with even $\mathcal{E}_1$, answering conjecture (3) of Andrews and Newman. In the published version he further connects the counting to Kostka polynomials and to level-one $\widehat{\mathfrak{sl}}_2$ character (branching-function) formulae.

The concluding remarks of the paper state two explicit hopes: that a more explicit bijective proof of the main equidistribution will be found (the existing proof matches generating functions analytically), and that a partition statistic with a more succinct definition - comparable to the three-line, notation-free definitions of rank and crank, using only the parts rather than diagrammatic procedures - will be found that reproduces the same integer sequence $p_{2,2}(n)$. A 2026 follow-up by Miyazawa and Takagi constructs explicit energy-preserving bijections between the sqrank/rerank classes and certain highest-weight paths in the crystal graphs of the level-one $U_q(\widehat{\mathfrak{sl}}_2)$ crystals $B(\Lambda_0)$ and $B(\Lambda_1)$ (a crystal is the colored directed graph encoding the combinatorial skeleton of a quantum-group representation; Kashiwara operators move along its arrows), but the equality between the number of such paths and the number of partitions in the corresponding mex class is still deduced from the analytic branching-function identity, not by a bijection.

Problem Statement

The common objective is to give the Takagi-Andrews-Newman equidistribution between minimal-excludant classes and the sqrank/rerank statistics a direct, succinct, purely combinatorial foundation, replacing the analytic generating-function proof. Concretely, the problem asks for the following linked deliverables.

(1) Direct bijective proof of the equidistribution. For all $n,r\in\mathbb{Z}_{\geq 0}$, construct an explicit weight-preserving bijection between the set of partitions $\lambda\vdash n$ with $\mathrm{mex}_{2,1}(\lambda)=2r+1$ and the set of partitions $\lambda\vdash n$ with $\mathrm{sqrank}(\lambda)=r$, and likewise between the set of partitions $\lambda\vdash n$ with $\mathrm{mex}_{2,2}(\lambda)=2r+2$ and the set of partitions $\lambda\vdash n$ with $\mathrm{rerank}(\lambda)=r$ (a single uniform construction covering both cases is welcome). The proof of bijectivity must be direct: the construction itself must establish the equinumerosity, rather than composing bijections whose connection to the mex classes rests on an analytic identity such as the branching-function/character equality through which the known crystal-path bijections (Miyazawa-Takagi 2026) are currently linked to the mex classes. Recall that the mex classes have an elementary structure - a partition has $\mathrm{mex}_{2,1}(\lambda)=2r+1$ exactly when $1,3,\dots,2r-1$ are parts of $\lambda$ and $2r+1$ is not, and $\mathrm{mex}_{2,2}(\lambda)=2r+2$ exactly when $2,4,\dots,2r$ are parts and $2r+2$ is not - so the target is a concrete bijection between multiplicity-defined families and the statistics defined by Durfee decomposition, rim-hook stripping, and Frobenius coordinates.

(2) Succinct statistic for $p_{2,2}(n)$. Find a partition statistic $s$ whose definition has brevity comparable to the classical rank and crank - statable in a few lines in terms of the parts and their multiplicities, without iterative Ferrers-diagram surgery - such that, for every $n\geq 0$, either finely $\#\{\lambda\vdash n: s(\lambda)=r\}=p^{(r)}_{2,2}(n)$ for all $r$, or at least coarsely $\#\{\lambda\vdash n: s(\lambda)\ \mathrm{even}\}=p_{2,2}(n)$; a comparably simple closed identity for $p_{2,2}(n)$, paralleling $p_{2,1}(n)=p_e(n)$, would equally answer this part. A proof of the counting identity is required. The statistic $s$ (and any such identity) must be defined independently of the mex class being counted: its definition may not refer to $\mathrm{mex}_{2,2}$, or to any statistic determined by $\mathrm{mex}_{2,2}$, so that the answer is a genuinely new parts-and-multiplicities invariant rather than a re-encoding of the class that $p_{2,2}(n)$ counts.

A complete resolution establishes the common objective by delivering both (1) and (2), each with proof; delivering either one alone constitutes substantial, independently reviewable partial progress.

References

1. Taichiro Takagi, A polynomial bosonic form of statistical configuration sums and the odd/even minimal excludant in integer partitions, Annals of Combinatorics (2026), DOI 10.1007/s00026-025-00792-9, arXiv:2412.19503, https://arxiv.org/abs/2412.19503
2. George E. Andrews and David Newman, The minimal excludant in integer partitions, Journal of Integer Sequences 23 (2020), Article 20.2.3, https://cs.uwaterloo.ca/journals/JIS/VOL23/Andrews/andrews5.html
3. Shane Chern, q-Log-concavity and q-unimodality of Gaussian polynomials and a problem of Andrews and Newman, Proceedings of the Japan Academy, Series A, Mathematical Sciences 99 (2023), pp. 33-36, DOI 10.3792/pjaa.99.007, https://doi.org/10.3792/pjaa.99.007
4. Sota Miyazawa and Taichiro Takagi, Novel energy preserving bijections between affine crystals for $U_q(\widehat{\mathfrak{sl}}_2)$ and integer partitions, arXiv:2605.30806 (2026), https://arxiv.org/abs/2605.30806
5. Bernard L. S. Lin and Han Liu, Identities and Inequalities Involving the Crank and Minimal Excludant of Integer Partitions, Annals of Combinatorics (2025), DOI 10.1007/s00026-025-00773-y, https://doi.org/10.1007/s00026-025-00773-y
6. George E. Andrews and Brian Hopkins, Extending Andrews and Newman's refinement of the crank-mex theorem, arXiv:2511.19698 (2025), https://arxiv.org/abs/2511.19698
7. Nargish Punia, On $k$-color mex-related partition functions, arXiv:2606.25698 (2026), https://arxiv.org/abs/2606.25698
8. Subhash Chand Bhoria, Pramod Eyyunni and Subhrangsu Santra, On the number of missing integers in partitions, arXiv:2604.12557 (2026), https://arxiv.org/abs/2604.12557
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10. Xiangyu Ding and Lisa Hui Sun, A combinatorial proof for the positivity of the normalized Jacobi triple product tails, arXiv:2606.27507 (2026), https://arxiv.org/abs/2606.27507