181 research outputs found
Counting descents, rises, and levels, with prescribed first element, in words
Recently, Kitaev and Remmel [Classifying descents according to parity, Annals
of Combinatorics, to appear 2007] refined the well-known permutation statistic
``descent'' by fixing parity of one of the descent's numbers. Results in that
paper were extended and generalized in several ways. In this paper, we shall
fix a set partition of the natural numbers , , and we study
the distribution of descents, levels, and rises according to whether the first
letter of the descent, rise, or level lies in over the set of words over
the alphabet . In particular, we refine and generalize some of the results
in [Counting occurrences of some subword patterns, Discrete Mathematics and
Theoretical Computer Science 6 (2003), 001-012.].Comment: 20 pages, sections 3 and 4 are adde
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
Enumeration of Dumont permutations avoiding certain four-letter patterns
In this paper, we enumerate Dumont permutations of the fourth kind avoiding
or containing certain permutations of length 4. We also conjecture a
Wilf-equivalence of two 4-letter patterns on Dumont permutations of the first
kind
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