Some formulas for the restricted r-Lah numbers

Shattuck, Mark (2018) Some formulas for the restricted r-Lah numbers. Annales Mathematicae et Informaticae, 49. pp. 123-140. ISSN 1787-6117

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Abstract

The r-Lah numbers, which we denote here by `(r)(n, k), enumerate partitions of an (n+r)-element set into k+r contents-ordered blocks in which the smallest r elements belong to distinct blocks. In this paper, we consider a restricted version `(r) m (n, k) of the r-Lah numbers in which block cardinalities are at most m. We establish several combinatorial identities for `(r) m (n, k) and obtain as limiting cases for large m analogous formulas for `(r)(n, k). Some of these formulas correspond to previously established results for `(r)(n, k), while others are apparently new also in the r-Lah case. Some generating function formulas are derived as a consequence and we conclude by considering a polynomial generalization of `(r) m (n, k) which arises as a joint distribution for two statistics defined on restricted r-Lah distributions. Keywords: restricted Lah numbers, polynomial generalization, r-Lah numbers, combinatorial identities MSC: 11B73, 05A19, 05A18

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