Enumerating Some Stable Partitions Involving Stirling and r-Stirling Numbers of the Second Kind

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Abstract

The coefficient of the chromatic polynomial counts the number of partitions of the vertex set of a simple and finite graph G into k independent vertex sets, equivalently, it gives the number of proper colorings of G with exactly k colors subject to some constraints. In this work, we study this invariant, we establish new formulas in this context for some families of graphs and we treat some specific cases as Thorn graphs. Consequently, we derive identities for the classical Stirling numbers of the second kind, besides that, this gives rise to new explicit formulae for the r-Stirling numbers of the second kind.

Keywords

Stirling numbers of the second kind r-stirling numbers of the second kind chromatic polynomial stables set partitions independent partitions generating functions graphical stirling numbers deletion–contraction principle thorn graphs 

Mathematics Subject Classification

Primary 11B73 05C15 Secondary 05C30 05A18 05A19 11B83 11B39 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • H. Belbachir
    • 1
  • M. A. Boutiche
    • 2
  • A. Medjerredine
    • 1
  1. 1.USTHB, Faculty of MathematicsRECITS LaboratoryBab EzzouarAlgeria
  2. 2.USTHB, Faculty of MathematicsLaROMaD LaboratoryBab EzzouarAlgeria

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