In this paper we consider weakly monotonic sequences of positive integers <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1282_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1,\lambda _2,\dots ,\lambda _k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <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> </mrow> </math></EquationSource> </InlineEquation> such that either <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1282_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _i\le i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo>≤</mo> <mi>i</mi> </mrow> </math></EquationSource> </InlineEquation> for all i (subdiagonal partitions) or <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13370_2025_1282_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _i\ge i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mi>i</mi> </msub> <mo>≥</mo> <mi>i</mi> </mrow> </math></EquationSource> </InlineEquation> for all i (superdiagonal partitions). We provide generating functions for these partitions with respect to size and number of parts, using various approaches involving multivariate generating functions, functional equations, area statistics on lattice paths, and bijections.

Subdiagonal and superdiagonal partitions

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2025年4月7日

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M. Archibald,
A. Blecher,
Sergi Elizalde,
A. Knopfmacher

摘要

In this paper we consider weakly monotonic sequences of positive integers $\lambda _1,\lambda _2,\dots ,\lambda _k$ $λ_{1}, λ_{2}, \dots, λ_{k}$ such that either $\lambda _i\le i$ $λ_{i} \leq i$ for all i (subdiagonal partitions) or $\lambda _i\ge i$ $λ_{i} \geq i$ for all i (superdiagonal partitions). We provide generating functions for these partitions with respect to size and number of parts, using various approaches involving multivariate generating functions, functional equations, area statistics on lattice paths, and bijections.