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<p>This paper explores two families of integer partition functions, <i>Q</i>(<i>d</i>;&#xa0;<i>n</i>) and <i>P</i>(<i>d</i>;&#xa0;<i>n</i>), which count partitions of <i>n</i> under certain restrictions on how many times each part can appear. The first function, <i>Q</i>(<i>d</i>;&#xa0;<i>n</i>), counts partitions where no part shows up <i>d</i> or more times. The second, <i>P</i>(<i>d</i>;&#xa0;<i>n</i>), involves a signed count of partitions where the number of times each part appears is either 0 or 1 modulo <i>d</i>. These functions generalize some well-known partition types—like distinct partitions—and turn out to have a surprisingly rich structure. We prove a collection of identities that tie these two functions together. This includes convolution formulas, mutual expansions, and an orthogonality relation. We also find recurrences and develop exact formulas for both <i>Q</i>(<i>d</i>;&#xa0;<i>n</i>) and <i>P</i>(<i>d</i>;&#xa0;<i>n</i>) using sums over partitions. Altogether, these results reveal a deeper algebraic and combinatorial picture behind partitions with multiplicity constraints.</p>

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Partitions with multiplicity constraints

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Mircea Merca

摘要

This paper explores two families of integer partition functions, Q(d; n) and P(d; n), which count partitions of n under certain restrictions on how many times each part can appear. The first function, Q(d; n), counts partitions where no part shows up d or more times. The second, P(d; n), involves a signed count of partitions where the number of times each part appears is either 0 or 1 modulo d. These functions generalize some well-known partition types—like distinct partitions—and turn out to have a surprisingly rich structure. We prove a collection of identities that tie these two functions together. This includes convolution formulas, mutual expansions, and an orthogonality relation. We also find recurrences and develop exact formulas for both Q(d; n) and P(d; n) using sums over partitions. Altogether, these results reveal a deeper algebraic and combinatorial picture behind partitions with multiplicity constraints.