Heaps and two exponential structures

Abstract

Take Q=(Q1,Q2,…) to be an exponential structure and M(n) to be the number of minimal elements of Qn where M(0)=1. Then a sequence of numbers {rn(Qn)}n≥1 is defined by the equation ∑n≥1rn(Qn)znn!M(n)=−log(∑n≥0(−1)nznn!M(n)). Let Q̄n denote the poset Qn with a 0ˆ adjoined and let 1ˆ denote the unique maximal element in the poset Qn. Furthermore, let μQn be the Möbius function on the poset Q̄n. Stanley proved that rn(Qn)=(−1)nμQn(0ˆ,1ˆ). This implies that the numbers rn(Qn) are integers. In this paper, we study the cases Qn=Πn(r) and Qn=Qn(r) where Πn(r) and Qn(r) are posets, respectively, of set partitions of [rn] whose block sizes are divisible by r and of r-partitions of [n]. In both cases we prove that rn(Πn(r)) and rn(Qn(r)) enumerate the pyramids by applying the Cartier–Foata monoid identity and further prove that rn(Πn(r)) is the generalized Euler number Ern−1 and that rn(Qn(2)) is the number of complete non-ambiguous trees of size 2n−1 by bijections. This gives a new proof of Welker’s theorem that rn(Πn(r))=Ern−1 and implies the construction of r-dimensional complete non-ambiguous trees. As a bonus of applying the theory of heaps, we establish a bijection between the set of complete non-ambiguous forests and the set of pairs of permutations with no common rise. This answers an open question raised by Aval et al.