scholarly journals Pattern avoidance for set partitions \`a la Klazar

2016 ◽  
Vol Vol. 18 no. 2, Permutation... (Permutation Patterns) ◽  
Author(s):  
Jonathan Bloom ◽  
Dan Saracino
Keyword(s):  
Pattern Avoidance ◽  
Set Partitions ◽  
Wilf Equivalence

In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| <|\Pi_n(\tau_2)|$ for all $n>k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$. Comment: 21 pages

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 Pattern Avoidance in Ascent Sequences

10.37236/713 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Paul Duncan ◽  
Einar Steingrímsson
Keyword(s):  
Growth Rates ◽  
Pattern Avoidance ◽  
Combinatorial Structures ◽  
Set Partitions ◽  
Combinatorial Objects ◽  
Ascent Sequences ◽  
Wilf Equivalence

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to $(2+2)$-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of conjectures related to all of these aspects.

scholarly journals Patterns in matchings and rook placements

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Jonathan Bloom ◽  
Sergi Elizalde
Keyword(s):  
Generating Functions ◽  
Direct Proof ◽  
Pattern Avoidance ◽  
Set Partitions ◽  
Rook Placements ◽  
International Audience ◽  
Wilf Equivalence ◽  
Diagram Representation ◽  
Ferrers Boards

International audience 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, unlike in the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which simplifies existing proofs by Backelin–West–Xin and Jelínek.

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 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.

scholarly journals Beyond Alternating Permutations: Pattern Avoidance in Young Diagrams and Tableaux

10.37236/3246 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Nihal Gowravaram ◽  
Ravi Jagadeesan
Keyword(s):  
Pattern Avoidance ◽  
General Context ◽  
Young Tableaux ◽  
Young Diagrams ◽  
Alternating Permutations ◽  
Equivalence Type ◽  
Shape Equivalence ◽  
Pattern Avoiding Permutations ◽  
Wilf Equivalence

We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young tableaux, that is, permutations that have $k-1$ ascents followed by a descent, followed by $k-1$ ascents, et cetera. We determine the relative sizes of the sets of pattern-avoiding $(k-1)$-ascent permutations in terms of the forbidden pattern. Furthermore, inequalities in the sizes of sets of pattern-avoiding permutations in this context arise from further extensions of shape-equivalence type enumerations. This paper is the first of a two-paper series presenting the work of Beyond alternating permutations: Pattern avoidance in Young diagrams and tableaux (arXiv:1301.6796v1). The second in the series is Ascent-descent Young diagrams and pattern avoidance in alternating permutations (by the second author, submitted).

scholarly journals Pattern Avoidance in Ordered Set Partitions

2014 ◽  
Vol 18 (3) ◽  
pp. 429-445 ◽  
Author(s):  
Anant Godbole ◽  
Adam Goyt ◽  
Jennifer Herdan ◽  
Lara Pudwell
Keyword(s):  
Ordered Set ◽  
Pattern Avoidance ◽  
Set Partitions ◽  
Ordered Set Partitions

scholarly journals Wilf-Equivalence on $k$-ary Words, Compositions, and Parking Functions

10.37236/147 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Pattern Avoidance ◽  
Equivalence Classes ◽  
Parking Functions ◽  
Arbitrary Size ◽  
Wilf Equivalence ◽  
Full Classification

In this paper, we study pattern-avoidance in the set of words over the alphabet $[k]$. We say that a word $w\in[k]^n$ contains a pattern $\tau\in[\ell]^m$, if $w$ contains a subsequence order-isomorphic to $\tau$. This notion generalizes pattern-avoidance in permutations. We determine all the Wilf-equivalence classes of word patterns of length at most six. We also consider analogous problems within the set of integer compositions and the set of parking functions, which may both be regarded as special types of words, and which contain all permutations. In both these restricted settings, we determine the equivalence classes of all patterns of length at most five. As it turns out, the full classification of these short patterns can be obtained with only a few general bijective arguments, which are applicable to patterns of arbitrary size.

scholarly journals Pattern avoidance in partial permutations (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Anders Claesson ◽  
Vít Jelínek ◽  
Eva Jelínková ◽  
Sergey Kitaev
Keyword(s):  
Equivalence Class ◽  
Arbitrary Number ◽  
Pattern Avoidance ◽  
Permutation Patterns ◽  
Nous Montrons ◽  
International Audience ◽  
Wilf Equivalence ◽  
Partial Words

International audience Motivated by the concept of partial words, we introduce an analogous concept of partial permutations. A $\textit{partial permutation of length n with k holes}$ is a sequence of symbols $\pi = \pi_1 \pi_2 \cdots \pi_n$ in which each of the symbols from the set $\{1,2,\ldots,n-k\}$ 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 $n \geq k \geq 1$. Nous introduisons un concept de permutations partielles. $\textit{Une permutation partielle de longueur n avec k trous}$ est une suite finie de symboles $\pi = \pi_1 \pi_2 \cdots \pi_n$ dans laquelle chaque nombre de l'ensemble $\{1,2,\ldots,n-k\}$ apparaît précisément une fois, tandis que les $k$ autres symboles de $\pi$ sont des "trous''. Nous introduisons l'étude des permutations partielles à motifs exclus et nous montrons que la plupart des résultats sur l'équivalence de Wilf peuvent être généralisés aux permutations partielles avec un nombre arbitraire de trous. De plus, nous montrons que les permutations de Baxter d'une longueur donnée $k$ forment une classe d'équivalence du type Wilf par rapport aux permutations partielles avec $(k-2)$ trous. Enfin, nous présentons l'énumération des permutations partielles de longueur $n$ avec $k$ trous qui évitent un motif de longueur $\ell \leq 4$, pour chaque $n \geq k \geq 1$.

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