scholarly journals A Congruence for the Number of Alternating Permutations

2021 ◽  
Vol 33 (1) ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Alternating Permutations

A Note on Alternating Permutations

2007 ◽  
Vol 114 (5) ◽  
pp. 437-440 ◽  
Author(s):  
Anthony Mendes
Keyword(s):  
Alternating Permutations

scholarly journals A q-enumeration of alternating permutations

2010 ◽  
Vol 31 (7) ◽  
pp. 1892-1906 ◽  
Author(s):  
Matthieu Josuat-Vergès
Keyword(s):  
Alternating Permutations

Generating alternating permutations lexicographically

1990 ◽  
Vol 30 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Bruce Bauslaugh ◽  
Frank Ruskey
Keyword(s):  
Alternating Permutations

scholarly journals On pattern avoiding alternating permutations

2014 ◽  
Vol 40 ◽  
pp. 11-25 ◽  
Author(s):  
Joanna N. Chen ◽  
William Y.C. Chen ◽  
Robin D.P. Zhou
Keyword(s):  
Alternating Permutations

scholarly journals Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

10.37236/2001 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Petter Brändén ◽  
Anders Claesson
Keyword(s):  
Fixed Points ◽  
Relative Position ◽  
Permutation Statistics ◽  
Permutation Patterns ◽  
Alternating Permutations ◽  
Major Index ◽  
Permutation Pattern ◽  
New Type ◽  
Permutation Statistic ◽  
Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

scholarly journals Pattern avoidance in alternating permutations and tableaux (extended abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Young Tableaux ◽  
Alternating Permutations ◽  
Special Cases ◽  
Nous Montrons ◽  
Insertion Algorithm ◽  
Generating Trees ◽  
International Audience ◽  
Standard Young Tableaux

International audience We give bijective proofs of pattern-avoidance results for a class of permutations generalizing alternating permutations. The bijections employed include a modified form of the RSK insertion algorithm and recursive bijections based on generating trees. As special cases, we show that the sets $A_{2n}(1234)$ and $A_{2n}(2143)$ are in bijection with standard Young tableaux of shape $\langle 3^n \rangle$. Alternating permutations may be viewed as the reading words of standard Young tableaux of a certain skew shape. In the last section of the paper, we study pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape $\lambda / \mu$ whose reading words avoid $213$ is a natural $\mu$-analogue of the Catalan numbers. Similar results for the patterns $132$, $231$ and $312$. Nous présentons des preuves bijectives de résultats pour une classe de permutations à motifs exclus qui généralisent les permutations alternantes. Les bijections utilisées reposent sur une modification de l'algorithme d'insertion "RSK" et des bijections récursives basées sur des arbres de génération. Comme cas particuliers, nous montrons que les ensembles $A_{2n}(1234)$ et $A_{2n}(2143)$ sont en bijection avec les tableaux standards de Young de la forme $\langle 3^n \rangle$. Une permutation alternante peut être considérée comme le mot de lecture de certain skew tableau. Dans la dernière section de l'article, nous étudions l'évitement des motifs dans les mots de lecture de skew tableaux généraux. Nous montrons bijectivement que le nombre de tableaux standards de forme $\lambda / \mu$ dont les mots de lecture évitent $213$ est un $\mu$-analogue naturel des nombres de Catalan. Des résultats analogues sont valables pour les motifs $132$, $231$ et $312$.

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