| 1. |
- Aas, Erik, et al.
(författare)
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Limiting directions for random walks in classical affine Weyl groups
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.
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| 2. |
- Aas, Erik, et al.
(författare)
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Limiting Directions for Random Walks in Classical Affine Weyl Groups
- 2021
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Ingår i: International mathematics research notices. - : Oxford University Press (OUP). - 1073-7928 .- 1687-0247. ; 2023:4, s. 3092-3137
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Tidskriftsartikel (refereegranskat)abstract
- Let W be a finite Weyl group and (W) over tilde the corresponding affine Weyl group. A random element of (W) over tilde can be obtained as a reduced random walk on the alcoves of (W) over tilde. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of vertical bar W vertical bar many directions. We compute these directions when W is B-n, C-n, and D-n, and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types B and C in the affirmative and for type D in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process (TASEP) called the D*-TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.
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| 3. |
- Aas, E., et al.
(författare)
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The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps
- 2019
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Ingår i: Journal of Physics A. - : Institute of Physics Publishing (IOPP). - 1751-8113 .- 1751-8121. ; 52:35
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Tidskriftsartikel (refereegranskat)abstract
- We consider a finite one-dimensional totally asymmetric simple exclusion process with four types of particles, {1, 0, 1, }, in contact with reservoirs. Particles of species 0 can neither enter nor exit the lattice, and those of species are constrained to lie at the first and last site. Particles of species 1 enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species 1 enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram.
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| 4. |
- Alexandersson, Per, et al.
(författare)
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Refined Catalan and Narayana cyclic sieving
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of “ears”, non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.
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| 5. |
- Alexandersson, Per, et al.
(författare)
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Refined Catalan and Narayana cyclic sieving
- 2021
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Ingår i: Combinatorial Theory. - : California Digital Library (CDL). - 2766-1334. ; 1:0
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Tidskriftsartikel (refereegranskat)abstract
- We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B . Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.
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| 6. |
- Alexandersson, Per, et al.
(författare)
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The cyclic sieving phenomenon on circular Dyck paths
- 2019
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Ingår i: The Electronic Journal of Combinatorics. - : ELECTRONIC JOURNAL OF COMBINATORICS. - 1097-1440 .- 1077-8926. ; 26:4
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Tidskriftsartikel (refereegranskat)abstract
- We give a q-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this q-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Mobius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.
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| 7. |
- Linusson, Svante, et al.
(författare)
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New properties of the Edelman-Greene bijection
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Edelman and Greene constructed a correspondence between reduced words of the reverse permutation and standard Young tableaux. We prove that for any reduced word the shape of the region of the insertion tableau containing the smallest possible entries evolves exactly as the upper-left component of the permutation's (Rothe) diagram. Properties of the Edelman-Greene bijection restricted to 132-avoiding and 2143-avoiding permutations are presented. We also consider the Edelman-Greene bijection applied to non-reduced words.
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| 8. |
- Linusson, Svante, et al.
(författare)
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On random shifted standard Young tableaux and 132-avoiding sorting networks
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is determined, with the integers in the SYT as heights. This implies via properties of the Edelman-Greene bijection results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency.
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| 9. |
- Linusson, Svante, et al.
(författare)
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On random shifted standard Young tableaux and 132-avoiding sorting networks
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is determined, with the integers in the SYT as heights. This implies via properties of the Edelman-Greene bijection results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency.
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| 10. |
- Linusson, Svante, et al.
(författare)
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On random shifted standard Young tableaux and 132-avoiding sorting networks
- 2020
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Ingår i: Algebraic Combinatorics. - : Cellule MathDoc/CEDRAM. - 2589-5486. ; 3:6, s. 1231-1258
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Tidskriftsartikel (refereegranskat)abstract
- We study shifted standard Young tableaux (SYT). The limiting surface of uniformly random shifted SYT of staircase shape is determined, with the integers in the SYT as heights. This implies via properties of the Edelman-Greene bijection results about random 132-avoiding sorting networks, including limit shapes for trajectories and intermediate permutations. Moreover, the expected number of adjacencies in SYT is considered. It is shown that on average each row and each column of a shifted SYT of staircase shape contains precisely one adjacency.
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