Abstract
We derive the asymptotics of certain combinatorial numbers defined on multi-sets when the number of sets tends to infinity but the sizes of the sets remain fixed. This includes the asymptotics of generalized derangements, numbers related to k-partite graphs, and exponentially weighted derangements. The asymptotics use integral and sum representations of the numbers involved. We also explore the combinatorial implications of the asymptotic results. In fact we first derive general asymptotic formulas for integrals and sums of certain types and then we specialize them to study the asymptotics of the combinatorial numbers.
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Communicated by Charles Micchelli.
This research is supported by the Research Grants Council of Hong Kong under contract # 101410. The research of Mourad E. H. Ismail is also supported by the NPST Program of King Saud University, Project Number 10-MAT1293-02, and a grant from King Saud University, Kingdom of Saudi Arabia.
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Ismail, M.E.H., Simeonov, P. Asymptotics of generalized derangements. Adv Comput Math 39, 101–127 (2013). https://doi.org/10.1007/s10444-011-9271-7
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DOI: https://doi.org/10.1007/s10444-011-9271-7
Keywords
- Weighted derangements
- k-partite graphs
- Laguerre polynomials
- Hermite polynomials
- Meixner polynomials
- Laplace asymptotic method