The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts

Abstract

We interpret the second difference of the sequence of the number of strict partitions, for n≥5, as the sequence of the strict partitions of n with at least three parts, the three largest parts consecutive, and the smallest part at least two. The name butterfly describes both a sequence's interpretation and a bijection between subsets of strict partitions. Using cyclotomic polynomials, we compute generating function identities of the butterfly sequence and related sequences both as infinite products and as series filtered by the number of parts. We offer a merging and splitting construction of the butterfly sequence as a sequence of partitions with odd parts larger or equal to 3, and we interpret the butterfly sequence as a sequence of generalized pentagonal, pentagonal with domino, and non-pentagonal butterfly partitions. Euler's Pentagonal Number Theorem and a similar specialization of the Jacobi Triple Product lead to recursive algorithms to compute the butterfly sequence and related sequences.