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| 2. |
- Aas, Erik, et al.
(author)
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Limiting directions for random walks in classical affine Weyl groups
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Other publication (other academic/artistic)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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| 3. |
- Alexandersson, Per, et al.
(author)
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Refined Catalan and Narayana cyclic sieving
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Other publication (other academic/artistic)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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| 4. |
- Linusson, Svante, et al.
(author)
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New properties of the Edelman-Greene bijection
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Other publication (other academic/artistic)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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| 5. |
- Linusson, Svante, et al.
(author)
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On random shifted standard Young tableaux and 132-avoiding sorting networks
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Other publication (other academic/artistic)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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| 6. |
- Linusson, Svante, et al.
(author)
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On random shifted standard Young tableaux and 132-avoiding sorting networks
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Other publication (other academic/artistic)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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| 7. |
- Linusson, Svante, et al.
(author)
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On the Edges of Characteristic Imset Polytopes
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Other publication (other academic/artistic)abstract
- The edges of the characteristic imset polytope, CIMp, were recently shown to have strong connections to causal discovery as many algorithms could be interpreted as greedy restricted edge-walks, even though only a strict subset of the edges are known. To better understand the general edge structure of the polytope we describe the edge structure of faces with a clear combinatorial interpretation: for any undirected graph G we have the face CIMG, the convex hull of the characteristic imsets of DAGs with skeleton G. We give a full edge-description of CIMG when G is a tree, leading to interesting connections to other polytopes. In particular the well-studied stable set polytope can be recovered as a face of CIMG when G is a tree. Building on this connection we are also able to give a description of all edges of CIMG when G is a cycle, suggesting possible inroads for generalization. We then introduce an algorithm for learning directed trees from data, utilizing our newly discovered edges, that outperforms classical methods on simulated Gaussian data.
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| 8. |
- Linusson, Svante, et al.
(author)
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Rhombus Criterion and the Chordal Graph Polytope
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Other publication (other academic/artistic)abstract
- The purpose of this paper is twofold. We investigate a simple necessary condition, called the rhombus criterion, for two vertices in a polytope not to form an edge and show that in many examples of 0/1-polytopes it is also sufficient. We explain how also when this is not the case, the criterion can give a good algorithm for determining the edges of high-dimenional polytopes.In particular we study the Chordal graph polytope, which arises in the theory of causality and is an important example of a characteristic imset polytope. We prove that, asymptotically, for almost all pairs of vertices the rhombus criterion holds. We conjecture it to hold for all pairs of vertices.
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