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

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 $2r+2$.

💡 Research Summary

The paper addresses a question raised by Andrews and Newman concerning the existence of a partition statistic that reproduces the same integer sequence as the even minimal excludant (mex) function. Building on the notion of the minimal excludant (\operatorname{mex}_{A,a}(\lambda)), which for a partition (\lambda) returns the smallest positive integer congruent to (a) modulo (A) that does not appear among the parts of (\lambda), the authors focus on the case (A=2). In this setting the “odd minimal excludant’’ takes values (2r+1) and the “even minimal excludant’’ takes values (2r+2). The sequences
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