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<p>In this paper, we count the total number of hooks of length two in all odd partitions of <i>n</i> and all distinct partitions of <i>n</i> with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of <i>n</i> in the presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester’s map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.</p>

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Finite analogs of partition bias related to hook length two and a variant of Sylvester’s map

  • 期刊论文

  • 发表日期：

    2025年7月18日

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  • Alexander Berkovich,
  • Aritram Dhar

摘要

In this paper, we count the total number of hooks of length two in all odd partitions of n and all distinct partitions of n with a bound on the largest part of the partitions. We generalize inequalities of Ballantine, Burson, Craig, Folsom and Wen by showing there is a bias in the number of hooks of length two in all odd partitions over all distinct partitions of n in the presence of a bound on the largest part. To establish such a bias, we use a variant of Sylvester’s map. Then, we conjecture a similar finite bias for a weighted count of hooks of length two and prove it when we remove the bound on the largest part.