93 research outputs found
An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences
We introduce an algorithmic approach based on generating tree method for
enumerating the inversion sequences with various pattern-avoidance
restrictions. For a given set of patterns, we propose an algorithm that outputs
either an accurate description of the succession rules of the corresponding
generating tree or an ansatz. By using this approach, we determine the
generating trees for the pattern-classes ,
, , ,
and . Then we use the kernel method, obtain generating functions
of each class, and find enumerating formulas. Lin and Yan studied the
classification of the Wilf-equivalences for inversion sequences avoiding pairs
of length-three patterns and showed that there are 48 Wilf classes among 78
pairs. In this paper, we solve six open cases for such pattern classes.Comment: 20 pages, 2 figure
Inversion sequences avoiding pairs of patterns
The enumeration of inversion sequences avoiding a single pattern was
initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck
independently. Their work has sparked various investigations of generalized
patterns in inversion sequences, including patterns of relation triples by
Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular
patterns by Lin and Yan. In this paper, we carried out the systematic study of
inversion sequences avoiding two patterns of length . Our enumerative
results establish further connections to the OEIS sequences and some classical
combinatorial objects, such as restricted permutations, weighted ordered trees
and set partitions. Since patterns of relation triples are some special
multiple patterns of length , our results complement the work by Martinez
and Savage. In particular, one of their conjectures regarding the enumeration
of -avoiding inversion sequences is solved
Isomorphisms between pattern classes
Isomorphisms p between pattern classes A and B are considered. It is shown
that, if p is not a symmetry of the entire set of permutations, then, to within
symmetry, A is a subset of one a small set of pattern classes whose structure,
including their enumeration, is determined.Comment: 11 page
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
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