70 research outputs found
Some generalizations of a simion-schmidt bijection
In 1985 Simion and Schmidt gave a constructive bijection 4 from the set of all length
(n ā 1) binary strings having no two consecutive is to the set of all length n permutations
avoiding all patterns in {123, 132,213).
In this paper we generalize 4 to an inject ive function from {O, i}n_1 to the set S, of all
length n permutations and derive from it four bijections : P ā* Q where P Ƨ {O, i}?_1
and Q C S,. The doniaiiis are sets of restricted binary strings and the codomains are sets of
pattern-avoiding permutations. As a particular case we retrieve the original Simion-Schmidt
bij ect ion.
We also show that the l)ijections obtained are actually combinatorial isomorphisms, i.e.,
closenessāpreserving bijections. Three of them have known Cray codes and generating alg
orithms for their domains and we present similar results for each corresponding codomain,
under the appropriate combinatorial isomorphism
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
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3
Vincular and covincular patterns are generalizations of classical patterns
allowing restrictions on the indices and values of the occurrences in a
permutation. In this paper we study the integer sequences arising as the
enumerations of permutations simultaneously avoiding a vincular and a
covincular pattern, both of length 3, with at most one restriction. We see
familiar sequences, such as the Catalan and Motzkin numbers, but also some
previously unknown sequences which have close links to other combinatorial
objects such as lattice paths and integer partitions. Where possible we include
a generating function for the enumeration. One of the cases considered settles
a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We
also give an alternative proof of the classic result that permutations avoiding
123 are counted by the Catalan numbers.Comment: 24 pages, 11 figures, 2 table
Place-difference-value patterns: A generalization of generalized permutation and word patterns
Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson
introduced the notion of a "generalized permutation pattern" (GP) which
generalizes the concept of "classical" permutation pattern introduced by Knuth
in 1969. The invention of GPs led to a large number of publications related to
properties of these patterns in permutations and words. Since the work of
Babson and Steingrimsson, several further generalizations of permutation
patterns have appeared in the literature, each bringing a new set of
permutation or word pattern problems and often new connections with other
combinatorial objects and disciplines. For example, Bousquet-Melou et al.
introduced a new type of permutation pattern that allowed them to relate
permutation patterns theory to the theory of partially ordered sets.
In this paper we introduce yet another, more general definition of a pattern,
called place-difference-value patterns (PDVP) that covers all of the most
common definitions of permutation and/or word patterns that have occurred in
the literature. PDVPs provide many new ways to develop the theory of patterns
in permutations and words. We shall give several examples of PDVPs in both
permutations and words that cannot be described in terms of any other pattern
conditions that have been introduced previously. Finally, we raise several
bijective questions linking our patterns to other combinatorial objects.Comment: 18 pages, 2 figures, 1 tabl
Permutations Restricted by Two Distinct Patterns of Length Three
Define to be the number of permutations on letters which avoid
all patterns in the set and contain each pattern in the multiset
exactly once. In this paper we enumerate and
for all . The
results for follow from two papers by Mansour and
Vainshtein.Comment: 15 pages, some relevant reference brought to my attention (see
section 4
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