Enumeration of labelled threshold graphs and a theorem of frobenius involving eulerian polynomials |
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Authors: | Janet Simpson Beissinger Uri N. Peled |
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Affiliation: | (1) Department of Mathematics, Statistics, and Computer Science, University of Ilinois at Chicago, Box 4348, 60680 Chicago, Il, USA |
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Abstract: | We enumerate labelled threshold graphs by the number of vertices, the number of isolated vertices, and the number of distinct vertex-degrees and we give the exact asymptotics for the number of labelled threshold graphs withn vertices. We obtain the appropriate generating function and point out a combinatorial interpretation relating its coefficients to the Stirling numbers of the second kind. We use these results to derive a new proof of a theorem of Frobenius expressing the Eulerian polynomials in terms of the Stirling numbers. |
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