Let NT(r, m, n) denote the total number of parts in all partitions of n whose rank is congruent to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$r \pmod {m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M_{\omega }(r,m,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>ω</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the total number of ones in partitions of n whose crank is congruent to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$r \pmod {m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In 2023, Xuan, Yao, and Zhou established several Andrews–Beck type congruences for NT(r, m, n) and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_{\omega }(r,m,n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>ω</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> using generating functions and posed the problem of finding combinatorial proofs. In this paper, we present such combinatorial proofs via Ferrers diagrams.

Combinatorial proofs for certain Andrews–Beck type congruences in Beck’s partition statistics

期刊论文
发表日期：

2026年4月22日

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Imdadul Hussain,
Suparno Ghoshal,
Arijit Jana

摘要

Let NT(r, m, n) denote the total number of parts in all partitions of n whose rank is congruent to $r \pmod {m}$ $r (mod m)$ , and let $M_{\omega }(r,m,n)$ $M_{ω} (r, m, n)$ denote the total number of ones in partitions of n whose crank is congruent to $r \pmod {m}$ $r (mod m)$ . In 2023, Xuan, Yao, and Zhou established several Andrews–Beck type congruences for NT(r, m, n) and $M_{\omega }(r,m,n)$ $M_{ω} (r, m, n)$ using generating functions and posed the problem of finding combinatorial proofs. In this paper, we present such combinatorial proofs via Ferrers diagrams.