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arXiv · 2412.19503

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

Taichiro Takagi
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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 $2r+2$.

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Taichiro Takagi. 2024-12-27. A polynomial bosonic form of statistical configuration sums and the odd/even minimal excludant in integer partitions. https://doi.org/10.1007/s00026-025-00792-9

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