195 research outputs found
Simple marked mesh patterns
In this paper we begin the first systematic study of distributions of simple
marked mesh patterns. Mesh patterns were introduced recently by Br\"and\'en and
Claesson in connection with permutation statistics. We provide explicit
generating functions in several general cases, and develop recursions to
compute the numbers in question in some other cases. Certain -analogues are
discussed. Moreover, we consider two modifications of the notion of a marked
mesh pattern and provide enumerative results for them.Comment: 27 page
The 1-box pattern on pattern avoiding permutations
This paper is continuation of the study of the 1-box pattern in permutations
introduced by the authors in \cite{kitrem4}. We derive a two-variable
generating function for the distribution of this pattern on 132-avoiding
permutations, and then study some of its coefficients providing a link to the
Fibonacci numbers. We also find the number of separable permutations with two
and three occurrences of the 1-box pattern
Row-strict quasisymmetric Schur functions
Haglund, Luoto, Mason, and van Willigenburg introduced a basis for
quasisymmetric functions called the quasisymmetric Schur function basis,
generated combinatorially through fillings of composition diagrams in much the
same way as Schur functions are generated through reverse column-strict
tableaux. We introduce a new basis for quasisymmetric functions called the
row-strict quasisymmetric Schur function basis, generated combinatorially
through fillings of composition diagrams in much the same way as Schur
functions are generated through row-strict tableaux. We describe the
relationship between this new basis and other known bases for quasisymmetric
functions, as well as its relationship to Schur polynomials. We obtain a
refinement of the omega transform operator as a result of these relationships.Comment: 17 pages, 11 figure
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
Generating functions for Wilf equivalence under generalized factor order
Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on
words comprised of letters from a partially ordered set by
setting if there is a subword of of the same length as
such that the -th character of is greater than or equal to the -th
character of for all . This subword is called an embedding of
into . For the case where is the positive integers with the usual
ordering, they defined the weight of a word to be
, and the corresponding weight
generating function . They then
defined two words and to be Wilf equivalent, denoted , if
and only if . They also defined the related generating
function where
is the set of all words such that the only embedding of
into is a suffix of , and showed that if and only if
. We continue this study by giving an explicit formula for
if factors into a weakly increasing word followed by a weakly
decreasing word. We use this formula as an aid to classify Wilf equivalence for
all words of length 3. We also show that coefficients of related generating
functions are well-known sequences in several special cases. Finally, we
discuss a conjecture that if then and must be
rearrangements, and the stronger conjecture that there also must be a
weight-preserving bijection such
that is a rearrangement of for all .Comment: 23 page
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