718 research outputs found

    Identically Distributed Pairs of Partition Statistics

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    We show that many theorems which assert that two kinds of partitions of the same integer nn are equinumerous are actually special cases of a much stronger form of equality. We show that in fact there correspond partition statistics XX and YY that have identical distribution functions. The method is an extension of the principle of sieve-equivalence, and it yields simple criteria under which we can infer this identity of distribution functions

    Pattern avoidance in compositions and multiset permutations

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    We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer compositions of n that avoid a given pattern of length 3 and we show that the answer is the same for all such patterns. We also show that the number of multiset permutations that avoid a given three-letter pattern is the same for all such patterns, thereby extending and refining earlier results of Albert, Aldred et al., and by Atkinson, Walker and Linton. Further, the number of permutations of a multiset S, with a_i copies of i for i = 1, ..., k, that avoid a given permutation pattern in S_3 is a symmetric function of the a_i's, and we will give here a bijective proof of this fact first for the pattern (123), and then for all patterns in S_3 by using a recently discovered bijection of Amy N. Myers.Comment: 8 pages, no figur

    Computing the distribution of the maximum in balls-and-boxes problems, with application to clusters of disease cases

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    We present a rapid method for the exact calculation of the cumulative distribution function of the maximum of multinomially distributed random variables. The method runs in time O(mn)O(mn), where mm is the desired maximum and nn is the number of variables. We apply the method to the analysis of two situations where an apparent clustering of cases of a disease in some locality has raised the possibility that the disease might be communicable, and this possibility has been discussed in the recent literature. We conclude that one of these clusters may be explained on purely random grounds, whereas the other may not