A rank of partitions with overline designated summands

2020/04/07 by Hao, Robert. X. J., Shen, Erin Y. Y., Zang, Wenston J. T.
#05A17 #05A20 #11P83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.03122

Abstract

Andrews, Lewis and Lovejoy introduced the partition function PD(n) as the number of partitions of n with designated summands. In a recent work, Lin studied a partition function PD_t(n) which counts the number of tagged parts over all the partitions of n with designated summands. He proved that PD_t(3n+2) is divisible by 3. In this paper, we first introduce a structure named partitions with overline designated summands, which is counted by PD_t(n). We then define a generalized rank of partitions with overline designated summands and give a combinatorial interpretation of the congruence for PD_t(3n+2).

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