Permutations and Pairs of Dyck Paths

Pattern Avoidance in Permutations: Linear and Cyclic Orders

The Electronic Journal of Combinatorics ◽

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.

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Bruhat Order on Fixed-Point-Free Involutions in the Symmetric Group

The Electronic Journal of Combinatorics ◽

10.37236/3861 ◽

2014 ◽

Vol 21 (2) ◽

Author(s):

Matthew Watson

Keyword(s):

Fixed Point ◽

Symmetric Group ◽

Bruhat Order ◽

Binary Trees ◽

Combinatorial Proof ◽

Combinatorial Identity ◽

Structural Description ◽

Rook Placements ◽

Dyck Paths ◽

New Interpretation

We provide a structural description of Bruhat order on the set $F_{2n}$ of fixed-point-free involutions in the symetric group $S_{2n}$ which yields a combinatorial proof of a combinatorial identity that is an expansion of its rank-generating function. The decomposition is accomplished via a natural poset congruence, which yields a new interpretation and proof of a combinatorial identity that counts the number of rook placements on the Ferrers boards lying under all Dyck paths of a given length $2n$. Additionally, this result extends naturally to prove new combinatorial identities that sum over other Catalan objects: 312-avoiding permutations, plane forests, and binary trees.

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Extending the parking space

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2325 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Andrew Berget ◽

Brendon Rhoades

Keyword(s):

Representation Theory ◽

Symmetric Group ◽

Algebraic Combinatorics ◽

Parking Space ◽

Parking Functions ◽

Dyck Paths ◽

International Audience ◽

Special Case

International audience The action of the symmetric group $S_n$ on the set $\mathrm{Park}_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $\mathrm{Park}_n$ extends to an action of $S_{n+1}$. More precisely, we construct a graded $S_{n+1}$-module $V_n$ such that the restriction of $V_n$ to $S_n$ is isomorphic to $\mathrm{Park}_n$. We describe the $S_n$-Frobenius characters of the module $V_n$ in all degrees and describe the $S_{n+1}$-Frobenius characters of $V_n$ in extreme degrees. We give a bivariate generalization $V_n^{(\ell, m)}$ of our module $V_n$ whose representation theory is governed by a bivariate generalization of Dyck paths. A Fuss generalization of our results is a special case of this bivariate generalization. L’action du groupe symétrique $S_n$ sur l’ensemble $\mathrm{Park}_n$ des fonctions de stationnement de longueur $n$ a reçu beaucoup d’attention dans la combinatoire algébrique. Nous démontrons que l’action de $S_n$ sur $\mathrm{Park}_n$ s’étend à une action de $S_{n+1}$. Plus précisément, nous construisons un gradué $S_{n+1}$-module $V_n$ telles que la restriction de $S_n$ est isomorphe à $\mathrm{Park}_n$. Nous décrivons la $S_n$-Frobenius caractères des modules $V_n$ à tous les degrés et décrivent le $S_{n+1}$-Frobenius caractères de $V_n$ en degrés extrêmes. Nous donnons une généralisation bivariée $V_n^{(\ell, m)}$ de notre module $V_n$ dont la représentation théorie est régie par une généralisation bivariée des chemins de Dyck. Une généralisation Fuss de nos résultats est un cas particulier de cette généralisation bivariée.

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Representations of the Infinite Symmetric Group

10.1017/cbo9781316798577 ◽

2016 ◽

Cited By ~ 3

Author(s):

Alexei Borodin ◽

Grigori Olshanski

Keyword(s):

Symmetric Group ◽

Infinite Symmetric Group

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The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

International Mathematics Research Notices ◽

10.1093/imrn/rnaa290 ◽

2020 ◽

Author(s):

Heather M Russell ◽

Julianna Tymoczko

Keyword(s):

Symmetric Group ◽

Partial Order ◽

Transition Matrix ◽

Group Representation ◽

Jones Polynomial ◽

Basis Matrix ◽

Combinatorial Structures ◽

Quantum Invariants ◽

One Dimensional ◽

Link Diagrams

Abstract Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

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