2023/03/24 by David Galvin, Galvin, David, John Engbers +3
Mathematics · #05A15 #05A18 #26C15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2303.14057
openalex publication_date 2023/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The reciprocal of e-x has a power series about 0 in which all coefficients are non-negative. Gessel [Reciprocals of exponential polynomials and permutation enumeration, Australas. J. Combin., 74, 2019] considered truncates of the power series of e-x, i.e. polynomials of the form ∑n=0r (-1)n(xn)/(n!), and established combinatorially that the reciprocal of the truncate has a power series with all coefficients non-negative precisely when r is odd. Here we extend Gessel's observations to arbitrary ``thinned exponential series''. To be precise, let A ⊆ \1,3,5,…\ and B ⊆ \2,4,6,…\, and consider the series 1-∑a ∈ A (xa)/(a!) + ∑b ∈ B (xb)/(b!). We consider conditions on A and B that ensure that the reciprocal series has all coefficients non-negative. We give combinatorial proofs for a large set of conditions, including whenever 1 ∈ A and the endpoints of the maximal consecutive intervals in A ∪ B are odd integers. In particular, the coefficients in the reciprocal series can be interpreted as ordered set partitions of [n] with block size restrictions, or in terms of permutations with restricted lengths of maximally increasing runs, suitably weighted.