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)].


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
Full Text: Link


[1] E. Barnard. “The canonical join complex”.Electron. J. Combin.26.1 (2019), Paper 1.24, 25 pp. Link. · Zbl 07032096
[2] A. E. Brouwer and A. Schrijver.On the period of an operator, defined on antichains. Mathematisch Centrum Afdeling Zuivere Wiskunde ZW 24/74. Mathematisch Centrum, Amsterdam, 1974, pp. i+13.
[3] P. J. Cameron and D. G. Fon-Der-Flaass. “Orbits of antichains revisited”.European J. Combin. 16.6 (1995), pp. 545-554.Link. · Zbl 0831.06001
[4] M. Chan, S. Haddadan, S. Hopkins, and L. Moci. “The expected jaggedness of order ideals”.Forum Math. Sigma5(2017), Art. e9, 27 pp.Link. · Zbl 1358.05313
[5] M. Chan, A. López Martín, N. Pflueger, and M. Teixidor i Bigas. “Genera of Brill-Noether curves and staircase paths in Young tableaux”.Trans. Amer. Math. Soc.370.5 (2018), pp. 3405- 3439.Link. · Zbl 1380.05007
[6] S. Hopkins. “The CDE property for skew vexillary permutations”.J. Combin. Theory Ser. A 168(2019), pp. 164-218.Link. · Zbl 1421.05004
[7] J. Propp and T. Roby. “Homomesy in products of two chains”.Electron. J. Combin.22.3 (2015), Paper 3.4, 29 pp.Link. · Zbl 1319.05151
[8] N. Reading. “Noncrossing arc diagrams and canonical join representations”.SIAM J. Discrete Math.29.2 (2015), pp. 736-750.Link. · Zbl 1314.05015
[9] V. Reiner, B. E. Tenner, and A. Yong. “Poset edge densities, nearly reduced words, and barely set-valued tableaux”.J. Combin. Theory Ser. A158(2018), pp. 66-125.Link. · Zbl 1391.05269
[10] J. Striker. “The toggle group, homomesy, and the Razumov-Stroganov correspondence”. Electron. J. Combin.22.2 (2015), Paper 2.57, 17 pp.Link. · Zbl 1319.05015
[11] J. Striker and N. Williams. “Promotion and rowmotion”.European J. Combin.33.8 (2012), pp. 1919-1942.Link. · Zbl 1260.06004
[12] H.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.