Summary: Take \(\mathsf{Q} = (\mathsf{Q}_1, \mathsf{Q}_2, \ldots)\) to be an exponential structure and \(M(n)\) to be the number of minimal elements of \(\mathsf{Q}_n\) where \(M(0) = 1\). Then a sequence of numbers \(\{r_n(\mathsf{Q}_n) \}_{n \geq 1}\) is defined by the equation \(\sum_{n \geq 1} r_n(\mathsf{Q}_n) \frac{z^n}{n! M(n)} = - \log(\sum_{n \geq 0}(- 1)^n \frac{z^n}{n! M(n)}) \text{.}\) Let \(\overline{\mathsf{Q}}_n\) denote the poset \(\mathsf{Q}_n\) with a \(\hat{0}\) adjoined and let \(\hat{1}\) denote the unique maximal element in the poset \(\mathsf{Q}_n\). Furthermore, let \(\mu_{\mathsf{Q}_n}\) be the Möbius function on the poset \(\overline{\mathsf{Q}}_n\). Stanley proved that \(r_n(\mathsf{Q}_n) = (- 1)^n \mu_{\mathsf{Q}_n}(\hat{0}, \hat{1})\). This implies that the numbers \(r_n(\mathsf{Q}_n)\) are integers.
In this paper, we study the cases \(\mathsf{Q}_n = \Pi_n^{(r)}\) and \(\mathsf{Q}_n = \mathsf{Q}_n^{(r)}\) where \(\Pi_n^{(r)}\) and \(\mathsf{Q}_n^{(r)}\) are posets, respectively, of set partitions of \([r n]\) whose block sizes are divisible by \(r\) and of \(r\)-partitions of \([n]\). In both cases we prove that \(r_n(\Pi_n^{(r)})\) and \(r_n(\mathsf{Q}_n^{(r)})\) enumerate the pyramids by applying the Cartier-Foata monoid identity and further prove that \(r_n(\Pi_n^{(r)})\) is the generalized Euler number \(E_{r n - 1}\) and that \(r_n(\mathsf{Q}_n^{(2)})\) is the number of complete non-ambiguous trees of size \(2 n - 1\) by bijections. This gives a new proof of Welker’s theorem that \(r_n(\Pi_n^{(r)}) = E_{r n - 1}\) 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 J.-C. Aval et al. [Adv. Appl. Math. 56, 78–108 (2014; Zbl 1300.05127)].