scholarly journals Descents and des-Wilf equivalence of permutations avoiding certain nonclassical patterns

2019 ◽  
Vol 12 (4) ◽  
pp. 549-563
Author(s):  
Caden Bielawa ◽  
Robert Davis ◽  
Daniel Greeson ◽  
Qinhan Zhou
Keyword(s):  
Wilf Equivalence

scholarly journals A General Theory of Wilf-Equivalence for Catalan Structures

10.37236/5629 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Michael Albert ◽  
Mathilde Bouvel
Keyword(s):  
General Theory ◽  
Equivalence Relation ◽  
Generating Functions ◽  
Asymptotic Estimate ◽  
Equivalence Classes ◽  
Catalan Number ◽  
Combinatorial Structures ◽  
Dyck Paths ◽  
Significant Interest ◽  
Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

scholarly journals Wilf equivalence relations for consecutive patterns

2018 ◽  
Vol 99 ◽  
pp. 134-157 ◽  
Author(s):  
Tim Dwyer ◽  
Sergi Elizalde
Keyword(s):  
Equivalence Relations ◽  
Wilf Equivalence

scholarly journals Wilf-equivalence for singleton classes

2007 ◽  
Vol 38 (2) ◽  
pp. 133-148 ◽  
Author(s):  
Jörgen Backelin ◽  
Julian West ◽  
Guoce Xin
Keyword(s):  
Wilf Equivalence

scholarly journals Rook and Wilf equivalence of integer partitions

2018 ◽  
Vol 71 ◽  
pp. 246-267 ◽  
Author(s):  
Jonathan Bloom ◽  
Dan Saracino
Keyword(s):  
Integer Partitions ◽  
Wilf Equivalence

scholarly journals Pattern Avoidance in Partial Permutations

10.37236/512 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Wilf Equivalence ◽  
Partial Words

Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A partial permutation of length $n$ with $k$ holes is a sequence of symbols $\pi=\pi_1\pi_2\dotsb\pi_n$ in which each of the symbols from the set $\{1,2,\dotsc,n-k\}$ appears exactly once, while the remaining $k$ symbols of $\pi$ are "holes". We introduce pattern-avoidance in partial permutations and prove that most of the previous results on Wilf equivalence of permutation patterns can be extended to partial permutations with an arbitrary number of holes. We also show that Baxter permutations of a given length $k$ correspond to a Wilf-type equivalence class with respect to partial permutations with $(k-2)$ holes. Lastly, we enumerate the partial permutations of length $n$ with $k$ holes avoiding a given pattern of length at most four, for each $n\ge k\ge 1$.

scholarly journals Pattern Avoidance in Matchings and Partitions

10.37236/2976 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Fixed Points ◽  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Set Partitions ◽  
Bijective Proof ◽  
Rook Placements ◽  
Wilf Equivalence ◽  
Diagram Representation

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize $3$-crossings and $3$-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards.We enumerate $312$-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the $321$-avoiding (i.e., $3$-noncrossing) case. Our approach provides a more direct proof of a formula of Bóna for the number of $1342$-avoiding permutations. We also give a bijective proof of the shape-Wilf-equivalence of the patterns $321$ and $213$ which greatly simplifies existing proofs by Backelin-West-Xin and Jelínek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

scholarly journals Two examples of unbalanced Wilf-equivalence

2015 ◽  
Vol 6 (1–2) ◽  
pp. 55-67
Author(s):  
Alexander Burstein ◽  
Jay Pantone
Keyword(s):  
Wilf Equivalence

scholarly journals Pattern Avoidance by Even Permutations

10.37236/2024 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Andrew Baxter ◽  
Aaron D. Jaggard
Keyword(s):  
Symmetric Group ◽  
Pattern Avoidance ◽  
Alternating Group ◽  
Wilf Equivalence

We study questions of even-Wilf-equivalence, the analogue of Wilf-equivalence when attention is restricted to pattern avoidance by permutations in the alternating group. Although some Wilf-equivalence results break when considering even-Wilf-equivalence analogues, we prove that other Wilf-equivalence results continue to hold in the even-Wilf-equivalence setting. In particular, we prove that $t(t-1)\cdots 321$ and $(t-1)(t-2)\cdots 21t$ are even-shape-Wilf-equivalent for odd $t$, paralleling a result (which held for all $t$) of Backelin, West, and Xin for shape-Wilf-equivalence. This allows us to classify the symmetric group $\mathcal{S}_{4}$, and to partially classify $\mathcal{S}_{5}$ and $\mathcal{S}_{6}$, according to even-Wilf-equivalence. As with transition to involution-Wilf-equivalence, some—but not all—of the classical Wilf-equivalence results are preserved when we make the transition to even-Wilf-equivalence.

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