scholarly journals Pattern Avoidance in Permutations: Linear and Cyclic Orders

10.37236/1690 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Antoine Vella
Keyword(s):  
Symmetric Group ◽  
Joint Distribution ◽  
Pattern Avoidance ◽  
Horizontal Edge ◽  
Ordered Sets ◽  
Dyck Paths ◽  
Euler Totient Function ◽  
Enumeration Problem ◽  
Counting Lemma ◽  
Horizontal And Vertical Edges

We generalize the notion of pattern avoidance to arbitrary functions on ordered sets, and consider specifically three scenarios for permutations: linear, cyclic and hybrid, the first one corresponding to classical permutation avoidance. The cyclic modification allows for circular shifts in the entries. Using two bijections, both ascribable to both Deutsch and Krattenthaler independently, we single out two geometrically significant classes of Dyck paths that correspond to two instances of simultaneous avoidance in the purely linear case, and to two distinct patterns in the hybrid case: non-decreasing Dyck paths (first considered by Barcucci et al.), and Dyck paths with at most one long vertical or horizontal edge. We derive a generating function counting Dyck paths by their number of low and high peaks, long horizontal and vertical edges, and what we call sinking steps. This translates into the joint distribution of fixed points, excedances, deficiencies, descents and inverse descents over 321-avoiding permutations. In particular we give an explicit formula for the number of 321-avoiding permutations with precisely $k$ descents, a problem recently brought up by Reifegerste. In both the hybrid and purely cyclic scenarios, we deal with the avoidance enumeration problem for all patterns of length up to 4. Simple Dyck paths also have a connection to the purely cyclic case; here the orbit-counting lemma gives a formula involving the Euler totient function and leads us to consider an interesting subgroup of the symmetric group.

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

scholarly journals Permutations and Pairs of Dyck Paths

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Marilena Barnabei ◽  
Flavio Bonetti ◽  
Matteo Silimbani
Keyword(s):  
Symmetric Group ◽  
Dyck Paths

We define a map v between the symmetric group Sn and the set of pairs of Dyck paths of semilength n. We show that the map v is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.

scholarly journals Ascent Sequences Avoiding Pairs of Patterns

10.37236/4479 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Andrew M. Baxter ◽  
Lara K. Pudwell
Keyword(s):  
Pattern Avoidance ◽  
Exact Enumeration ◽  
Combinatorial Objects ◽  
Dyck Paths ◽  
Generating Trees ◽  
Precise Bound ◽  
Ascent Sequences ◽  
Pattern Avoiding Permutations

Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and Steingrímsson. In this paper, we consider ascent sequences of length $n$ avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the Erdős-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound.

scholarly journals Bijections for lattice paths between two boundaries

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Sergi Elizalde ◽  
Martin Rubey
Keyword(s):  
Joint Distribution ◽  
Tutte Polynomial ◽  
Lattice Paths ◽  
Dyck Paths ◽  
International Audience ◽  
Pattern Avoiding Permutations

International audience We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations. On montre que, sur l'ensemble des chemins avec des pas $N=(0,1)$ et $E=(1,0)$ qui se trouvent entre deux chemins donnés $B$ et $T$, les deux statistiques"`nombre des pas $E$ en commun avec $B$" et "nombre des pas $E$ en commun avec $T$" ont une distribution conjointe symétrique. On donne une involution qui échange ces deux statistiques, préserve quelques autres paramètres additionnels, et admet une généralisation à des chemins avec des pas $S=(0, -1)$ dans des positions données. On montre aussi un autre résultat d'équidistribution similaire, lié au polynôme de Tutte d'un matroïde. Finalement, on étend les deux théorèmes à $k$-tuples de chemins entre deux frontières, et on donne quelques applications aux chemins de Dyck, en généralisant un résultat de Deutsch, et aux permutations avec des motifs exclus.

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 On the Joint Distribution of Descents and Signs of Permutations

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Jason Fulman ◽  
Gene B. Kim ◽  
Sangchul Lee ◽  
T. Kyle Petersen
Keyword(s):  
Symmetric Group ◽  
Central Limit ◽  
Generating Functions ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Coxeter Groups ◽  
Central Limit Theorems ◽  
Hyperoctahedral Group ◽  
Eulerian Numbers

We study the joint distribution of descents and sign for elements of the symmetric group and the hyperoctahedral group (Coxeter groups of types $A$ and $B$). For both groups, this has an application to riffle shuffling: for large decks of cards the sign is close to random after a single shuffle. In both groups, we derive generating functions for the Eulerian distribution refined according to sign, and use them to give two proofs of central limit theorems for positive and negative Eulerian numbers.

scholarly journals Transitive Factorizations in the Hyperoctahedral Group

2008 ◽  
Vol 60 (2) ◽  
pp. 297-312
Author(s):  
G. Bini ◽  
I. P. Goulden ◽  
D. M. Jackson
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Point Of View ◽  
Moduli Space Of Curves ◽  
Reflection Groups ◽  
Hurwitz Numbers ◽  
Hyperoctahedral Group ◽  
Geometric Point ◽  
Enumeration Problem

AbstractThe classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type A to other finite reflection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type B case that is studied here.

scholarly journals On some Euler-Mahonian Distributions

10.37236/6702 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Angela Carnevale
Keyword(s):  
Symmetric Group ◽  
Joint Distribution ◽  
Definition Of

We prove that the pair of statistics (des,maj) on multiset permutations is equidistributed with the pair (stc,inv) on certain quotients of the symmetric group. We define an analogue of the statistic stc on multiset permutations whose joint distribution with the inversions equals that of (des,maj). We extend the definition of the statistic stc to hyperoctahedral and even hyperoctahedral groups. These functions, together with the Coxeter length, are equidistributed with (ndes,nmaj) and (ddes,dmaj), respectively.

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