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The CDE property for skew vexillary permutations. (English) Zbl 1436.05003

Summary: We prove a conjecture of V. Reiner et al. [J. Comb. Theory, Ser. A 158, 66–125 (2018; Zbl 1391.05269)] which says that the initial weak order intervals corresponding to certain vexillary permutations have the coincidental down-degree expectations (CDE) property. Actually our theorem applies more generally to certain “skew vexillary” permutations (a notion we introduce), and shows that these posets are in fact “toggle CDE.” As a corollary we obtain a homomesy result for rowmotion acting on semidistributive lattices in the sense of E. Barnard [Electron. J. Comb. 26, No. 1, Research Paper P1.24, 25 p. (2019; Zbl 07032096)] and of H. Thomas and N. Williams [Proc. Lond. Math. Soc. (3) 119, No. 5, 1149–1178 (2019; Zbl 1459.06010)].

MSC:

05A05 Permutations, words, matrices
05E10 Combinatorial aspects of representation theory
20F55 Reflection and Coxeter groups (group-theoretic aspects)
06B15 Representation theory of lattices
06A07 Combinatorics of partially ordered sets
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References:

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