论 r 链最小和最大排除子的组合学

Pub Date : 2024-08-02 DOI:10.1016/j.disc.2024.114187
Subhash Chand Bhoria , Pramod Eyyunni , Runqiao Li
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引用次数: 0

摘要

分区的最小排除因子(mex)由格拉布纳(Grabner)和克诺普夫马赫(Knopfmacher)以 "最小间隙 "为名提出,最近由安德鲁斯(Andrews)和纽曼(Newman)重新提出。近年来,它与 Chern 首次提出的互补分区统计量最大排除因子(maximal excludant,maex)一起被广泛研究。在这些最新研究成果中,第一和第二作者与马吉一起提出并研究了 r 链最小不等式(r-chain mex),从而对欧拉经典分割定理以及安德鲁斯和纽曼的 sum-of-mex 特性进行了新的概括。在本文中,我们首先给出了关于 r 链 mex 的这两个结果的组合证明。然后,我们还通过分析和组合的方法,建立了前两位作者和马吉最近提出的 r 链最大不等式的相关同一性。
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On the combinatorics of r-chain minimal and maximal excludants

The minimal excludant (mex) of a partition was introduced by Grabner and Knopfmacher under the name ‘least gap’ and was recently revived by Andrews and Newman. It has been widely studied in recent years together with the complementary partition statistic maximal excludant (maex), first introduced by Chern. Among such recent works, the first and second authors along with Maji introduced and studied the r-chain minimal excludants (r-chain mex) which led to a new generalization of Euler's classical partition theorem and the sum-of-mex identity of Andrews and Newman. In this paper, we first give combinatorial proofs for these two results on r-chain mex. Then we also establish the associated identity for the r-chain maximal excludant, recently introduced by the first two authors and Maji, both analytically and combinatorially.

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