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<p>We study cylindric partitions with two-element profiles using MacMahon’s partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some <i>q</i>-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews–Gordon and Bressoud identities, which are companion to Foda–Quano’s refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.</p>

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A MacMahon analysis view of cylindric partitions

  • 期刊论文

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  • 发表日期：

    2025年9月22日

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  • Runqiao Li,
  • Ali K. Uncu

摘要

We study cylindric partitions with two-element profiles using MacMahon’s partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some q-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews–Gordon and Bressoud identities, which are companion to Foda–Quano’s refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.