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On the Probability that Certain Compositions Have the Same Number Of Parts
Miklós Bóna1 and Arnold Knopfmacher2
1Department of Mathematics, University of Florida, Gainesville FL 32611-8105, USA
bona@sas.upenn.edu
2School of Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, WITS 2050, South Africa
Arnold.Knopfmacher@wits.ac.za
Annals of Combinatorics 14 (3) pp.291-306 September, 2010
AMS Subject Classification: 05A05, 05A15, 05A16
Abstract:
We compute the asymptotic probability that two randomly selected compositions of n into parts equal to a or b have the same number of parts. In addition, we provide bijections in the case of parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets. Explicit algebraic generating functions and asymptotic probabilities are also computed in the case of pairs of compositions of n into parts at least d, for any fixed natural number d.
Keywords: composition, probability, lattice paths, fence poset, asymptotics

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