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A simple combinatorial interpretation of certain generalized Bell and Stirling numbers. (English) Zbl 1339.11033

Summary: In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that is relevant to the so-called boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpretation of these numbers. We show that by restricting the numbers somewhat (but still widely generalizing Bell and Stirling numbers), one can supply a much more natural combinatorial interpretation. In fact, we offer two different such interpretations, one in terms of graph colourings and another one in terms of certain labelled Eulerian digraphs.

MSC:

11B73 Bell and Stirling numbers
05A10 Factorials, binomial coefficients, combinatorial functions
05C15 Coloring of graphs and hypergraphs
05C45 Eulerian and Hamiltonian graphs

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References:

[1] Blasiak, P.; Flajolet, P., Combinatorial models of creation-annihilation, Sém. Lothar. Combin., 65 (2011), art. B65c. arXiv:1010.0354 [math.CO] · Zbl 1295.05062
[2] Blasiak, P.; Horzela, A.; Penson, K. A.; Solomon, A. I.; Duchamp, G. H.E., Combinatorics and Boson normal ordering: a gentle introduction, Amer. J. Phys., 75, 7, 639-646 (2007) · Zbl 1219.81165
[3] Blasiak, P.; Penson, K. A.; Solomon, A. I., The Boson normal ordering problem and generalized Bell numbers, Ann. Comb., 7, 2, 127-139 (2003), English · Zbl 1030.81004
[4] Blasiak, P.; Penson, K. A.; Solomon, A. I., The general boson normal ordering problem, Phys. Lett. A, 309, 3-4, 198-205 (2003) · Zbl 1009.81026
[5] Comtet, L., The art of finite and infinite expansions, (Advanced Combinatorics (1974), D. Reidel Publishing Co.: D. Reidel Publishing Co. Dordrecht)
[7] Méndez, M. A.; Blasiak, P.; Penson, K. A., Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem, J. Math. Phys., 46, 8 (2005), 083511 · Zbl 1110.81114
[8] The On-line Encyclopedia of Integer Sequences (2013), Published electronically at http://oeis.org · Zbl 1274.11001
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