<p>We introduce the subsum polynomial of a partition <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda =(\lambda _1, \lambda _2, \ldots , \lambda _k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{sp}(\lambda , x)=\prod _{i=1}^k(1+x^{\lambda _i})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>sp</mtext> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∏</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <msub> <mi>λ</mi> <mi>i</mi> </msub> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We study the sum of reciprocals of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{sp}(\lambda , x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>sp</mtext> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> over all partitions of <i>n</i>. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.</p>

Reciprocals of subsum polynomials

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2026年6月15日

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Cristina Ballantine,
George Beck,
Brooke Feigon,
Kathrin Maurischat

摘要

We introduce the subsum polynomial of a partition $\lambda =(\lambda _1, \lambda _2, \ldots , \lambda _k)$ $λ = (λ_{1}, λ_{2}, \dots, λ_{k})$ defined by $\textrm{sp}(\lambda , x)=\prod _{i=1}^k(1+x^{\lambda _i})$ $sp (λ, x) = \prod_{i = 1}^{k} (1 + x^{λ_{i}})$ . We study the sum of reciprocals of $\textrm{sp}(\lambda , x)$ $sp (λ, x)$ over all partitions of n. We prove arithmetic properties of related polynomials and offer connections to other combinatorial objects.