On the number of parts in all partitions enumerated by the Rogers-Ramanujan identities

2023/03/06 by Ballantine, Cristina, Folsom, Amanda
#05A17 #05A19 #11P84 #33D15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.03330

Abstract

The celebrated Rogers-Ramanujan identities equate the number of integer partitions of n (n∈\mathbb N₀) with parts congruent to ± 1 \pmod5 (respectively ± 2 \pmod5) and the number of partitions of n with super-distinct parts (respectively super-distinct parts greater than 1). In this paper, we establish companion identities to the Rogers-Ramanujan identities on the number of parts in all partitions of n of the aforementioned types, in the spirit of earlier work by Andrews and Beck on a partition identity of Euler.

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