Asymptotics of commuting ℓ -tuples in symmetric groups and log-concavity.

Pub Date : 2024-01-01 Epub Date: 2024-10-03 DOI:10.1007/s40993-024-00562-1

Kathrin Bringmann, Johann Franke, Bernhard Heim

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Abstract

Denote by $N_{ℓ} (n)$ the number of $ℓ$ -tuples of elements in the symmetric group $S_{n}$ with commuting components, normalized by the order of $S_{n}$ . In this paper, we prove asymptotic formulas for $N_{ℓ} (n)$ . In addition, general criteria for log-concavity are shown, which can be applied to $N_{ℓ} (n)$ among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form $c (a) c (b) > c (a + b)$ for certain families of sequences c(n).

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对称群中交换 ℓ -图元的渐近性和对数凹性

用 N ℓ ( n ) 表示对称群 S n 中具有交换成分的元素的 ℓ 元组数，以 S n 的阶归一化。本文证明了 N ℓ ( n ) 的渐近公式。此外，本文还展示了对数凹性的一般标准，这些标准可应用于 N ℓ ( n ) 及其他例子。此外，我们还得到了一个贝森罗特-奥诺（Bessenrodt-Ono）类型的定理，它给出了某些序列族 c(n) 的 c ( a ) c ( b ) > c ( a + b ) 的不等式。

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