Codara, Pietro; D’Antona, Ottavio M.; Hell, Pavol A simple combinatorial interpretation of certain generalized Bell and Stirling numbers. (English) Zbl 1339.11033 Discrete Math. 318, 53-57 (2014). 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. Cited in 6 Documents MSC: 11B73 Bell and Stirling numbers 05A10 Factorials, binomial coefficients, combinatorial functions 05C15 Coloring of graphs and hypergraphs 05C45 Eulerian and Hamiltonian graphs Keywords:generalized Bell number; generalized Stirling number; labelled Eulerian graph; graph colouring Software:OEIS PDFBibTeX XMLCite \textit{P. Codara} et al., Discrete Math. 318, 53--57 (2014; Zbl 1339.11033) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1). Triple factorial numbers (3*n-2)!!! with leading 1 added. Number of oriented multigraphs on n labeled arcs (without loops). Triangle of generalized Stirling numbers S_{2,2}(n,k) read by rows (n>=1, 2<=k<=2n). Triangle of generalized Stirling numbers S_{3,3}(n,k) read by rows (n>=1, 3<=k<=3n). Generalized Stirling2 array S_{4,4}(n,k). Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!. 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.