Enumeration of idempotent-sum subsequences in finite cyclic semigroups and smooth sequences

2025/05/01 by Wang, Guoqing, Zhao, Yang, Yi, Xingliang
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.00486

Abstract

The enumeration of zero-sum subsequences of a given sequence over finite cyclic groups is one classical topic, which starts from one question of P. Erdős. In this paper, we consider this problem in a more general setting -- finite cyclic semigroups. Let S be a finite cyclic semigroup. By e we denote the unique idempotent of the semigroup S. Let T be a sequence over the semigroup S, and let N(T; e) be the number of distinct subsequences of T with sum being the idempotent e. We obtain the lower bound for N(T; e) in terms of the length of T, and moreover, prove that T contains subsequences with some smooth-structure in case that N(T; e) is not large. Our result generalizes the theorem obtained by W. Gao [Discrete Math., 1994] on the enumeration of zero-sum subsequences over finite cyclic groups to the setting of semigroups.

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