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<p>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 non-negative integers <i>n</i>, we prove that the partitions of <i>n</i> on which sqrank or rerank takes on a particular value, say <i>r</i>, are equinumerous with the partitions of <i>n</i> on which the odd/even minimal exclutant takes on the corresponding value, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2r+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>r</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(2r+2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>r</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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A Polynomial Bosonic Form of Statistical Configuration Sums and the Odd/Even Minimal Excludant in Integer Partitions

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  • 发表日期：

    2026年2月6日

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  • Taichiro Takagi

摘要

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 non-negative 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, \(2r+1\) 2 r + 1 or \(2r+2\) 2 r + 2 .