Pattern avoiding permutations & rook placements

2014 ◽  
Author(s):  
Jonathan S. Bloom
Keyword(s):  
Rook Placements ◽  
Pattern Avoiding Permutations

Implicit Generation of Pattern-Avoiding Permutations by Using Permutation Decision Diagrams

2014 ◽  
Vol E97.A (6) ◽  
pp. 1171-1179
Author(s):  
Yuma INOUE ◽  
Takahisa TODA ◽  
Shin-ichi MINATO
Keyword(s):  
Decision Diagrams ◽  
Pattern Avoiding Permutations

scholarly journals On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

scholarly journals Restricted non-separable planar maps and some pattern avoiding permutations

2013 ◽  
Vol 161 (16-17) ◽  
pp. 2514-2526 ◽  
Author(s):  
Sergey Kitaev ◽  
Pavel Salimov ◽  
Christopher Severs ◽  
Henning Ulfarsson
Keyword(s):  
Planar Maps ◽  
Pattern Avoiding Permutations

scholarly journals Pattern-avoiding permutations and Brownian excursion, part II: fixed points

2016 ◽  
Vol 169 (1-2) ◽  
pp. 377-424 ◽  
Author(s):  
Christopher Hoffman ◽  
Douglas Rizzolo ◽  
Erik Slivken
Keyword(s):  
Fixed Points ◽  
Brownian Excursion ◽  
Pattern Avoiding Permutations

scholarly journals On the sub-permutations of pattern avoiding permutations

2014 ◽  
Vol 337 ◽  
pp. 127-141
Author(s):  
Filippo Disanto ◽  
Thomas Wiehe
Keyword(s):  
Pattern Avoiding Permutations

scholarly journals Aztec Diamonds and Baxter Permutations

10.37236/377 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Hal Canary
Keyword(s):  
Permutation Matrix ◽  
Enumerative Combinatorics ◽  
Alternating Sign Matrices ◽  
Domino Tilings ◽  
Permutation Matrices ◽  
Pattern Avoiding Permutations ◽  
The Relationship

We present a proof of a conjecture about the relationship between Baxter permutations and pairs of alternating sign matrices that are produced from domino tilings of Aztec diamonds. It is shown that a tiling corresponds to a pair of ASMs that are both permutation matrices if and only if the larger permutation matrix corresponds to a Baxter permutation. There has been a thriving literature on both pattern-avoiding permutations of various kinds [Baxter 1964, Dulucq and Guibert 1988] and tilings of regions using dominos or rhombuses as tiles [Elkies et al. 1992, Kuo 2004]. However, there have not as of yet been many links between these two areas of enumerative combinatorics. This paper gives one such link.

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 Encoding Labelled p-Riordan Graphs by Words and Pattern-Avoiding Permutations

2020 ◽  
Vol 37 (1) ◽  
pp. 139-149
Author(s):  
Kittitat Iamthong ◽  
Ji-Hwan Jung ◽  
Sergey Kitaev
Keyword(s):  
Pattern Avoiding Permutations

scholarly journals On bijections for pattern-avoiding permutations

2009 ◽  
Vol 116 (8) ◽  
pp. 1271-1284 ◽  
Author(s):  
Jonathan Bloom ◽  
Dan Saracino
Keyword(s):  
Pattern Avoiding Permutations

scholarly journals (2+2)-free posets, ascent sequences and pattern avoiding permutations

2010 ◽  
Vol 117 (7) ◽  
pp. 884-909 ◽  
Author(s):  
Mireille Bousquet-Mélou ◽  
Anders Claesson ◽  
Mark Dukes ◽  
Sergey Kitaev
Keyword(s):  
Ascent Sequences ◽  
Pattern Avoiding Permutations
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