110 research outputs found
Enumeration of super-strong Wilf equivalence classes of permutations in the generalized factor order
Super-strong Wilf equivalence classes of the symmetric group
on letters, with respect to the generalized factor order, were shown by
Hadjiloucas, Michos and Savvidou (2018) to be in bijection with pyramidal
sequences of consecutive differences. In this article we enumerate the latter
by giving recursive formulae in terms of a two-dimensional analogue of
non-interval permutations. As a by-product, we obtain a recursively defined set
of representatives of super-strong Wilf equivalence classes in . We also provide a connection between super-strong Wilf equivalence and
the geometric notion of shift equivalence---originally defined by Fidler,
Glasscock, Miceli, Pantone, and Xu (2018) for words---by showing that an
alternate way to characterize super-strong Wilf equivalence for permutations is
by keeping only rigid shifts in the definition of shift equivalence. This
allows us to fully describe shift equivalence classes for permutations of size
and enumerate them, answering the corresponding problem posed by Fidler,
Glasscock, Miceli, Pantone, and Xu (2018).Comment: 18 pages, 5 table
On super-strong Wilf equivalence classes of permutations
Super-strong Wilf equivalence is a type of Wilf equivalence on words that was originally introduced as strong Wilf equivalence by Kitaev et al. [Electron. J. Combin. 16(2)] in 2009. We provide a necessary and sufficient condition for two permutations in n letters to be super-strongly Wilf equivalent, using distances between letters within a permutation. Furthermore, we give a characterization of such equivalence classes via two-colored binary trees. This allows us to prove, in the case of super-strong Wilf equivalence, the conjecture stated in the same article by Kitaev et al. that the cardinality of each Wilf equivalence class is a power of 2
Wilf Equivalences and Stanley-Wilf Limits for Patterns in Rooted Labeled Forests
Building off recent work of Garg and Peng, we continue the investigation into
classical and consecutive pattern avoidance in rooted forests, resolving some
of their conjectures and questions and proving generalizations whenever
possible. Through extensions of the forest Simion-Schmidt bijection introduced
by Anders and Archer, we demonstrate a new family of forest-Wilf equivalences,
completing the classification of forest-Wilf equivalence classes for sets
consisting of a pattern of length 3 and a pattern of length at most . We
also find a new family of nontrivial c-forest-Wilf equivalences between single
patterns using the forest analogue of the Goulden-Jackson cluster method,
showing that a -fraction of patterns of length satisfy a
nontrivial c-forest-Wilf equivalence and that there are c-forest-Wilf
equivalence classes of patterns of length of exponential size.
Additionally, we consider a forest analogue of super-strong-c-Wilf equivalence,
introduced for permutations by Dwyer and Elizalde, showing that
super-strong-c-forest-Wilf equivalences are trivial by enumerating linear
extensions of forest cluster posets. Finally, we prove a forest analogue of the
Stanley-Wilf conjecture for avoiding a single pattern as well as certain other
sets of patterns. Our techniques are analytic, easily generalizing to different
types of pattern avoidance and allowing for computations of convergent lower
bounds of the forest Stanley-Wilf limit in the cases covered by our result. We
end with several open questions and directions for future research, including
some on the limit distributions of certain statistics of pattern-avoiding
forests.Comment: 53 pages, 19 figure
Consecutive Patterns in Inversion Sequences
An inversion sequence of length is an integer sequence such that for each .
Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of
patterns in inversion sequences, focusing on the enumeration of those that
avoid classical patterns of length 3. We initiate an analogous systematic study
of consecutive patterns in inversion sequences, namely patterns whose entries
are required to occur in adjacent positions. We enumerate inversion sequences
that avoid consecutive patterns of length 3, and generalize some results to
patterns of arbitrary length. Additionally, we study the notion of Wilf
equivalence of consecutive patterns in inversion sequences, as well as
generalizations of this notion analogous to those studied for permutation
patterns. We classify patterns of length up to 4 according to the corresponding
Wilf equivalence relations.Comment: Final version to appear in DMTC
Shift Equivalence in the Generalized Factor Order
We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given
Consecutive Pattern Containment and c-Wilf Equivalence
We derive linear recurrences for avoiding non-overlapping consecutive
patterns in both permutations and words and identify a necessary condition for
establishing c-Wilf-equivalence between two non-overlapping patterns.
Furthermore, we offer alternative, elementary proofs for several known results
in consecutive pattern containment that were previously demonstrated using
ideas from cluster algebra and analytical combinatorics. Lastly, we establish
new general bounds on the growth rates of consecutive pattern avoidance in
permutations
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