Generalized ordered set partitions. (English) Zbl 1444.05017
Summary: In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first \(r\) elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For this new combinatorial sequence we derive its exponential generating function, some recurrence relations, and combinatorial identities. We prove and present results using combinatorial arguments, generating functions, the symbolic method and Riordan arrays. For some specific cases we provide a combinatorial interpretation for the inverse matrix of the generalized Lah numbers by means of two families of posets.
MSC:
| 05A18 | Partitions of sets |
| 06A07 | Combinatorics of partially ordered sets |
| 05A15 | Exact enumeration problems, generating functions |
Online Encyclopedia of Integer Sequences:
Number of ”sets of lists”: number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.References:
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