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

TL;DR

This paper introduces new partition statistics, sqrank and rerank, and establishes their equinumerous relationship with odd/even minimal excludant in integer partitions, supported by polynomial bosonic forms of statistical sums.

Contribution

It presents novel partition statistics and connects them to minimal excludant concepts through polynomial bosonic forms and integrable cellular automata.

Findings

sqrank and rerank are equinumerous with odd/even minimal excludant values

Partitions with fixed sqrank or rerank correspond to those with specific minimal excludant values

The study links partition statistics to polynomial bosonic forms of statistical sums.

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 $2r+2$.