Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A polynomial bosonic form of statistical configuration sums and the odd/even minimal excludant in integer partitions

Taichiro Takagi

TL;DR

The paper introduces sqrank and rerank, two partition statistics derived from Ferrers diagrams via a Durfee-rectangle–driven procedure, and proves that partitions of $n$ with ${\rm sqrank}=r$ (resp. ${\rm rerank}=r$) are equinumerous with partitions of $n$ whose odd (resp. even) minimal excludant equals $2r{+}1$ (resp. $2r{+}2$). A central feature is a polynomial bosonic form for statistical configuration sums associated with an integrable cellular automaton, which underpins generating-function identities and energy-preserving bijections between restricted partitions, bit sequences, and pairs of partitions. The work connects these mex-related statistics to Gaussian polynomials, Kostka polynomials, and affine Lie algebra character theory, providing both analytic and combinatorial routes to the main equinumerosity result and answering Andrews–Newman’s question in this setting. It also yields corollaries such as a direct interpretation of $p_{2,2}(n)$ in terms of partitions with even mex and situates the findings within broader representation-theoretic frameworks. The paper closes with remarks on potential more succinct statistics and invites explicit bijective proofs to replace the analytic approach used for the main theorem.

Abstract

Inspired by the study of the minimal excludant in integer partitions by G.E. Andrews and D. Newman, we introduce a pair of new partition statistics, sqrank and rerank. They are related to a polynomial bosonic form of statistical configuration sums for an integrable cellular automaton. For all nonnegative integers $n$, we prove that the partitions of $n$ on which sqrank or rerank takes on a particular value, say $r$, are equinumerous with the partitions of $n$ on which the odd/even minimal exclutant takes on the corresponding value, $2r+1$ or $2r+2$.

A polynomial bosonic form of statistical configuration sums and the odd/even minimal excludant in integer partitions

TL;DR

The paper introduces sqrank and rerank, two partition statistics derived from Ferrers diagrams via a Durfee-rectangle–driven procedure, and proves that partitions of with (resp. ) are equinumerous with partitions of whose odd (resp. even) minimal excludant equals (resp. ). A central feature is a polynomial bosonic form for statistical configuration sums associated with an integrable cellular automaton, which underpins generating-function identities and energy-preserving bijections between restricted partitions, bit sequences, and pairs of partitions. The work connects these mex-related statistics to Gaussian polynomials, Kostka polynomials, and affine Lie algebra character theory, providing both analytic and combinatorial routes to the main equinumerosity result and answering Andrews–Newman’s question in this setting. It also yields corollaries such as a direct interpretation of in terms of partitions with even mex and situates the findings within broader representation-theoretic frameworks. The paper closes with remarks on potential more succinct statistics and invites explicit bijective proofs to replace the analytic approach used for the main theorem.

Abstract

Inspired by the study of the minimal excludant in integer partitions by G.E. Andrews and D. Newman, we introduce a pair of new partition statistics, sqrank and rerank. They are related to a polynomial bosonic form of statistical configuration sums for an integrable cellular automaton. For all nonnegative integers , we prove that the partitions of on which sqrank or rerank takes on a particular value, say , are equinumerous with the partitions of on which the odd/even minimal exclutant takes on the corresponding value, or .
Paper Structure (13 sections, 13 theorems, 91 equations, 6 figures, 2 tables)

This paper contains 13 sections, 13 theorems, 91 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and the main result
  3. Generating functions
  4. A polynomial bosonic form for statistical configuration sums
  5. A linear recursion relation associated with the Gaussian polynomials
  6. Canonical partition functions: statistical configuration sums over bit sequences
  7. An energy preserving bijection between restricted partitions and bit sequences
  8. An energy preserving bijection between partitions and pairs of partitions
  9. Proof of the main result
  10. Concluding remarks
  11. Relation to the Kostka polynomials
  12. Relation to the affine Lie algebra character formulae
  13. Relation to the configuration sums by Andrews-Baxter-Forrester

Key Result

Theorem 1

It holds that

Figures (6)

  • Figure 1: (left) The Ferrers diagram for partition $\lambda = (7,7,5,4,4,2,1).$; (right) Its decomposition into the hooks for recognizing its Frobenius representation $F(\lambda) =(6,5,2,0 \mid 6,4,2,1)$.
  • Figure 2: (a) The diagram of $\lambda$ and its Durfee square $D_0(\lambda)$ (shaded). (b) The rim hooks with arm length 3 that will be stripped away from the sub-diagram $A_0(\lambda)$. (c) The residual diagram $R_0(\lambda)$ and its diagonal node (shaded).
  • Figure 3: (a) The diagram of $\lambda$ and its Durfee rectangle $D_1(\lambda)$ (shaded). (b) The rim hooks with arm length 4 that will be stripped away from the sub-diagram $A_1(\lambda)$. (c) The residual diagram $R_1(\lambda)$ and its diagonal nodes (shaded), and its decomposition into the hooks.
  • Figure 4: A graphical representation of the path $S(\eta)$ for $\eta=01011001100100$ by line segments. We put a mark $\bullet$ between coordinate $i-1$ and $i$ if $\eta_i = 1$.
  • Figure 5: A graphical representation of maps $\tilde{f}_1$ and $\tilde{e}_1$. We have $\tilde{f}_1(01011001100100) = 01011001110100 \in \mathcal{H}^{(2)} (14,7)$ and $\tilde{e}_1(01011001110100) = 01011001100100 \in \mathcal{H}^{(1)} (14,6)$. The values of the energy coincide as $H(01011001100100) = H(01011001110100) = 1+3+7+11=22$.
  • ...and 1 more figures

Theorems & Definitions (39)

  • Definition 1
  • Example 1
  • Example 2
  • Theorem 1
  • Example 3
  • Corollary 2
  • Proposition 3
  • proof
  • Remark 1
  • Remark 2
  • ...and 29 more