4,443 research outputs found
Central limit theorems for patterns in multiset permutations and set partitions
We use the recently developed method of weighted dependency graphs to prove
central limit theorems for the number of occurrences of any fixed pattern in
multiset permutations and in set partitions. This generalizes results for
patterns of size 2 in both settings, obtained by Canfield, Janson and
Zeilberger and Chern, Diaconis, Kane and Rhoades, respectively.Comment: version 2 (52 pages) implements referee's suggestions and uses
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An extension of MacMahon's Equidistribution Theorem to ordered multiset partitions
A classical result of MacMahon states that inversion number and major index
have the same distribution over permutations of a given multiset. In this work
we prove a strengthening of this theorem originally conjectured by Haglund. Our
result can be seen as an equidistribution theorem over the ordered partitions
of a multiset into sets, which we call ordered multiset partitions. Our proof
is bijective and involves a new generalization of Carlitz's insertion method.
This generalization leads to a new extension of Macdonald polynomials for hook
shapes. We use our main theorem to show that these polynomials are symmetric
and we give their Schur expansion.Comment: An extended abstract of this work was presented at FPSAC 201
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
Permutations over cyclic groups
Generalizing a result in the theory of finite fields we prove that, apart
from a couple of exceptions that can be classified, for any elements
of the cyclic group of order , there is a permutation
such that
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