An integration of Euler's pentagonal partition

Abstract

A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler's recurrence based on the pentagonal numbers, but where the coefficients result from a discrete integration of Euler's coefficients. Both a bijective proof and one based on generating functions show the equivalence of the subject recurrences.Comment: 22 pages, 2 figures. The recurrence investigated in this paper is essentially that proposed in Exercise 5.2.3 of Igor Pak's "Partition bijections, a survey", Ramanujan J. 12 (2006), but casted in a different form and, perhaps more interestingly, endowed with a bijective proof which arises from a construction by induction on maximal part