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Restricted r-Stirling numbers and their combinatorial applications

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  • Bényi, Beáta
  • Méndez, Miguel
  • Ramírez, José L.
  • Wakhare, Tanay

Abstract

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.

Suggested Citation

  • Bényi, Beáta & Méndez, Miguel & Ramírez, José L. & Wakhare, Tanay, 2019. "Restricted r-Stirling numbers and their combinatorial applications," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 186-205.
  • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:186-205
    DOI: 10.1016/j.amc.2018.11.047
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    References listed on IDEAS

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    1. Takao Komatsu & Vichian Laohakosol & Kálmán Liptai, 2013. "A Generalization of Poly-Cauchy Numbers and Their Properties," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, December.
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