The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2

TL;DR

This paper explores the butterfly sequence derived from integer partitions with distinct parts, revealing its pentagonal structure, combinatorial identities, and connections to cyclotomic polynomials, with implications for generating functions and recursive algorithms.

Contribution

It introduces the butterfly sequence, interprets its combinatorial structure, and derives new generating function identities and recursive algorithms using cyclotomic polynomials and pentagonal numbers.

Findings

The butterfly sequence relates to partitions with specific constraints.

Generating functions for the sequence are expressed as infinite products and series.

Recursive algorithms for the sequence are developed using pentagonal numbers.

Abstract

Based on the author's previous work on the Jacobi identity for twisted relative vertex operator algebras and modules and on the generating function identities for affine Lie algebras, we interpret the second difference sequence of the sequence of the number of integer partitions with distinct parts (the strict partitions) 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 the sequence's interpretation and the underlying bijection between the set of strict partitions of positive integers m with the two largest parts consecutive, and a subset of the same kind of strict partitions of m+1. Using the cyclotomic polynomials 1-x and 1+x+x^2, we compute generating function identities o the butterfly sequence and related sequences both as infinite products and as series filtered by the number of parts of the corresponding partitions, and we see that the number of partitions of positive integers n with odd parts greater than or equal to 5 is the sum of three consecutive terms of the butterfly sequence. We also determine a subtler merging and splitting construction of the butterfly sequence as a sequence of some of the partitions with odd parts larger or equal to 3, and we offer a related detailed interpretation of the butterfly sequence as a sequence of what we define as generalized pentagonal, pentagonal with domino, and non-pentagonal butterfly partitions. Finally, Euler's Pentagonal Number Theorem and a slightly different specialization of the Jacobi triple product lead to recursive algorithms to compute the butterfly sequence and related sequences using pentagonal number sequences and the series of triangular powers.