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<p>In 1988, George Andrews and Frank Garvan discovered a crank for <i>p</i>(<i>n</i>), and provided its generating function. In 2020, Larry Rolen, Zack Tripp, and Ian Wagner generalized the crank generating function for <i>p</i>(<i>n</i>) in order to accommodate Ramanujan-like congruences for <i>k</i>-colored partitions by utilizing the theory of theta blocks developed by Gritsenko, Skorrupa, and Zagier. In this paper, we utilize some of the techniques used by Rolen, Tripp, and Wagner for crank generating functions in order to define a crank generating function for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_625_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\((k+j)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-colored partitions where <i>j</i> colors have distinct parts. We provide three infinite families of crank generating functions and conjecture a general crank generating function for such partitions. Further, we provide possible avenues that could be used in order to prove the general crank generating function. These results provide an alternative proof technique for some known congruences as well as a proof for some new congruences.</p>

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A proposed Crank for \((k+j)-\)colored partitions, with j colors having distinct parts

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Samuel Wilson

摘要

In 1988, George Andrews and Frank Garvan discovered a crank for p(n), and provided its generating function. In 2020, Larry Rolen, Zack Tripp, and Ian Wagner generalized the crank generating function for p(n) in order to accommodate Ramanujan-like congruences for k-colored partitions by utilizing the theory of theta blocks developed by Gritsenko, Skorrupa, and Zagier. In this paper, we utilize some of the techniques used by Rolen, Tripp, and Wagner for crank generating functions in order to define a crank generating function for \((k+j)\) ( k + j ) -colored partitions where j colors have distinct parts. We provide three infinite families of crank generating functions and conjecture a general crank generating function for such partitions. Further, we provide possible avenues that could be used in order to prove the general crank generating function. These results provide an alternative proof technique for some known congruences as well as a proof for some new congruences.