723 research outputs found
Identically Distributed Pairs of Partition Statistics
We show that many theorems which assert that two kinds of partitions of the
same integer are equinumerous are actually special cases of a much stronger
form of equality. We show that in fact there correspond partition statistics
and 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
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
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 , where is the desired maximum
and 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
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