s-Modular, s-congruent and s-duplicate partitions

2024/08/24 by Nadji, Mohammed L., Moussa, Ahmia
#05A17 #11P81 #11P83 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.13589

Abstract

In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) s-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to 0 or 1 modulo s, (2) s-congruent partitions, which generalize Sellers' partitions into parts not congruent to 2 modulo 4, and (3) s-duplicate partitions, of which the partitions having distinct odd parts and enumerated by the function \mypod(n) are a special case. In this vein, we generalize Alladi's series expansion for the product generating function of \mypod(n) and show that Andrews' generalization of Göllnitz-Gordon identities coincides with the number of partitions into parts simultaneously s-congruent and t-distinct (parts appearing fewer than t times).

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