Geodesic
Cyclic symmetry of the scaled simplex
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 225-246.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $\mathcal{Z}_{m}^{k}$ consist of the $m^{k}$ alcoves contained in the $m$-fold dilation of the fundamental alcove of the type $A_{k}$ affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order $k+1$, so does $\mathcal{Z}_{m}^{k}$. By bijectively exchanging the natural poset structure of $\mathcal{Z}_{m}^{k}$ for a natural cyclic action on a set of words, we prove that $(\mathcal{Z}_{m}^{k},\prod_{i=1}^{k} \frac{1-q^{mi}}{1-q^{i}},C_{k+1})$ exhibits the cyclic sieving phenomenon.
Classification : 05E05, 20C30, 05A10
Keywords: cyclic sieving phenomenon, affine permutations, type A alcove, core, abacus, Young's lattice 
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     title = {Cyclic symmetry of the scaled simplex},
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Thomas, Hugh; Williams, Nathan. Cyclic symmetry of the scaled simplex. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 225-246. http://geodesic.mathdoc.fr/item/JAC_2014__39_2_a9/