In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show connections between the different areas where poly-Bernoulli numbers and their relatives appear and give examples how the combinatorial methods can be used for deriving formulas between integer arrays.
We study set partitions with r distinguished elements and block sizes found in an arbitrary index set S. The enumeration of these (S, r)-partitions leads to the introduction of (S, r)-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the r-Stirling numbers. We also introduce the associated (S, r)-Bell and (S, r)-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For S with some extra structure, we show that the inverse of the (S, r)-Stirling matrix encodes the Möbius functions of two families of posets. Through several examples, we demonstrate that for some S the matrices and their inverses involve the enumeration sequences of several combinatorial objects. Further, we highlight how the (S, r)-Stirling numbers naturally arise in the enumeration of cliques and acyclic orientations of special graphs, underlining their ubiquity and importance. Finally, we introduce related (S, r) generalizations of the poly-Bernoulli and poly-Cauchy numbers, uniting many past works on generalized combinatorial sequences.
We construct a new bijection between the set of n × k 0-1 matrices with no three 1's forming a Γ configuration and the set of (n, k)-Callan sequences, a simple structure counted by poly-Bernoulli numbers. We give two applications of this result: We derive the generating function of Γ-free matrices, and we give a new bijective proof for an elegant result of Aval et al. that states that the number of complete non-ambiguous forests with n leaves is equal to the number of pairs of permutations of {1, . . . , n} with no common rise.
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