$$\sigma _o\text {mex}(n)$$ 和 $$\sigma _e\text {mex}(n)$$ 的算术性质和渐近公式

The Ramanujan Journal Pub Date : 2024-06-24 DOI:10.1007/s11139-024-00886-7

Rupam Barman, Gurinder Singh

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引用次数: 0

摘要

一个整数分割的最小不等式是分割中缺少的最小正整数。让 $\sigma _o\text {mex}(n)$ （或者， $\sigma _e\text {mex}(n)$ ）表示 n 的所有分区的奇数（或者，偶数）最小不等式之和。最近，巴鲁阿（Baruah）等人证明了这些分治函数 modulo 4 和 8 的一些同余式，并要求得到同样的渐近公式。在这篇文章中，我们找到了 $\sigma _o\text {mex}(n)$ 和 $\sigma _e\text {mex}(n)$ 的哈代-拉玛努扬式渐近公式。我们还证明了 $\sigma _o\text {mex}(n)$ 和 $\sigma _e\text {mex}(n)$ modulo 4 和 8 的一些无穷全等族。

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Arithmetic properties and asymptotic formulae for $$\sigma _o\text {mex}(n)$$ and $$\sigma _e\text {mex}(n)$$

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let $\sigma _o\text {mex}(n)$ (resp., $\sigma _e\text {mex}(n)$) denote the sum of odd (resp., even) minimal excludants over all the partitions of n. Recently, Baruah et al. proved a few congruences for these partition functions modulo 4 and 8, and asked for asymptotic formulae for the same. In this article, we find Hardy-Ramanujan type asymptotic formulae for both $\sigma _o\text {mex}(n)$ and $\sigma _e\text {mex}(n)$. We also prove some infinite families of congruences for $\sigma _o\text {mex}(n)$ and $\sigma _e\text {mex}(n)$ modulo 4 and 8

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The Ramanujan Journal

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