For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary q-series identities and generating function manipulations, along with mathematical induction: For all <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1111_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and all <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1111_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation><Equation ID="Equ10"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1111_Article_Equ10.gif" Format="GIF" Height="90" Rendition="HTML" Resolution="72" Type="Linedraw" Width="349" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} {{\,\textrm{pend}\,}}\left( 3^{2\alpha +1}n+\frac{17\cdot 3^{2\alpha }-1}{8}\right)&\equiv 0 \pmod {3}, \text { and} \\ {{\,\textrm{pond}\,}}\left( 3^{2\alpha +1}n+\frac{23\cdot 3^{2\alpha }+1}{8}\right)&\equiv 0 \pmod {3} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>pend</mtext> <mspace width="0.166667em" /> </mrow> <mfenced close=")" open="("> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>n</mi> <mo>+</mo> <mfrac> <mrow> <mn>17</mn> <mo>·</mo> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mn>8</mn> </mfrac> </mfenced> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>≡</mo> <mn>0</mn> <mspace width="10.0pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="0.333333em" /> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow /> <mrow> <mspace width="0.166667em" /> <mtext>pond</mtext> <mspace width="0.166667em" /> </mrow> <mfenced close=")" open="("> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>n</mi> <mo>+</mo> <mfrac> <mrow> <mn>23</mn> <mo>·</mo> <msup> <mn>3</mn> <mrow> <mn>2</mn> <mi>α</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> <mn>8</mn> </mfrac> </mfenced> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mo>≡</mo> <mn>0</mn> <mspace width="10.0pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1111_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{pend}\,}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>pend</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> counts the number of PEND partitions of weight n and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1111_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">${{\,\textrm{pond}\,}}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>pond</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> counts the number of POND partitions of weight n. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin–Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.

Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin–Lehner involution

期刊论文
出版模式：

开放获取
发表日期：

2025年5月23日

查看全文

James A. Sellers,
Nicolas Allen Smoot

摘要

For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary q-series identities and generating function manipulations, along with mathematical induction: For all $\alpha \ge 1$ $α \geq 1$ and all $n\ge 0,$ $n \geq 0,$ $\begin{aligned} {{\,\textrm{pend}\,}}\left( 3^{2\alpha +1}n+\frac{17\cdot 3^{2\alpha }-1}{8}\right)&\equiv 0 \pmod {3}, \text { and} \\ {{\,\textrm{pond}\,}}\left( 3^{2\alpha +1}n+\frac{23\cdot 3^{2\alpha }+1}{8}\right)&\equiv 0 \pmod {3} \end{aligned}$ $\begin{matrix} pend (3^{2 α + 1} n + \frac{17 \cdot 3^{2 α} - 1}{8}) & \equiv 0 (mod 3), and \\ pond (3^{2 α + 1} n + \frac{23 \cdot 3^{2 α} + 1}{8}) & \equiv 0 (mod 3) \end{matrix}$ where ${{\,\textrm{pend}\,}}(n)$ $pend (n)$ counts the number of PEND partitions of weight n and ${{\,\textrm{pond}\,}}(n)$ $pond (n)$ counts the number of POND partitions of weight n. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin–Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.