Combinatorics of diagrams of permutations

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:137

Article

Lewis, Joel Brewster¹  Morales, Alejandro H.² 

[1] Univ Minnesota, Minneapolis, MN 55455 USA

[2] Univ Calif Los Angeles, Los Angeles, CA USA

关键词: Permutation diagram;    Acyclic orientation;    Grassmannian;    q-Analogue;    Rook placement;    Le diagram;   

DOI  :  10.1016/j.jcta.2015.09.004

来源: Elsevier

【 摘 要 】

There are numerous combinatorial objects associated to a Grassmannian permutation w(lambda) that index cells of the totally nonnegative Grassmannian. We study several of these objects and their q-analogues in the case of permutations w that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of w, and fillings of a diagram of w are equinumerous for all permutations w. Second, we give a q-analogue of a result of Hultman-Linusson-Shareshian-Sjostrand by showing that under a certain pattern condition the Poincare polynomial for the Bruhat interval of w essentially counts invertible matrices over a finite field avoiding a diagram of w. In addition to our main results, we include at the end,a number of open questions. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free