***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001242 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. Given a Dyck path, there is an associated (directed) unit interval graph $\Gamma$. Consider the expansion $$G_\Gamma(x;q) = \sum_{\kappa: V(G) \to \mathbb{N}_+} x_\kappa q^{\mathrm{asc}(\kappa)}$$ using the notation by Alexandersson and Panova. The function $G_\Gamma(x;q)$ is a so called unicellular LLT polynomial, and a symmetric function. Consider the Schur expansion $$G_\Gamma(x;q+1) = \sum_{\lambda} c^\Gamma_\lambda(q) s_\lambda(x).$$ By a result by Haiman and Grojnowski, all $c^\Gamma_\lambda(q)$ have non-negative integer coefficients. Thus, $G_\Gamma(x;q+1)$ is the Frobenius image of some (graded) $S_n$-module. The total dimension of this $S_n$-module is $$D_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1)f^\lambda$$ where $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$. This statistic is $D_\Gamma$. ----------------------------------------------------------------------------- References: [1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles [[DOI:10.1016/j.disc.2018.09.001]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1,0] => 1 [1,0,1,0] => 2 [1,1,0,0] => 3 [1,0,1,0,1,0] => 6 [1,0,1,1,0,0] => 9 [1,1,0,0,1,0] => 9 [1,1,0,1,0,0] => 13 [1,1,1,0,0,0] => 21 [1,0,1,0,1,0,1,0] => 24 [1,0,1,0,1,1,0,0] => 36 [1,0,1,1,0,0,1,0] => 36 [1,0,1,1,0,1,0,0] => 52 [1,0,1,1,1,0,0,0] => 84 [1,1,0,0,1,0,1,0] => 36 [1,1,0,0,1,1,0,0] => 54 [1,1,0,1,0,0,1,0] => 52 [1,1,0,1,0,1,0,0] => 75 [1,1,0,1,1,0,0,0] => 117 [1,1,1,0,0,0,1,0] => 84 [1,1,1,0,0,1,0,0] => 117 [1,1,1,0,1,0,0,0] => 183 [1,1,1,1,0,0,0,0] => 315 [1,0,1,0,1,0,1,0,1,0] => 120 [1,0,1,0,1,0,1,1,0,0] => 180 [1,0,1,0,1,1,0,0,1,0] => 180 [1,0,1,0,1,1,0,1,0,0] => 260 [1,0,1,0,1,1,1,0,0,0] => 420 [1,0,1,1,0,0,1,0,1,0] => 180 [1,0,1,1,0,0,1,1,0,0] => 270 [1,0,1,1,0,1,0,0,1,0] => 260 [1,0,1,1,0,1,0,1,0,0] => 375 [1,0,1,1,0,1,1,0,0,0] => 585 [1,0,1,1,1,0,0,0,1,0] => 420 [1,0,1,1,1,0,0,1,0,0] => 585 [1,0,1,1,1,0,1,0,0,0] => 915 [1,0,1,1,1,1,0,0,0,0] => 1575 [1,1,0,0,1,0,1,0,1,0] => 180 [1,1,0,0,1,0,1,1,0,0] => 270 [1,1,0,0,1,1,0,0,1,0] => 270 [1,1,0,0,1,1,0,1,0,0] => 390 [1,1,0,0,1,1,1,0,0,0] => 630 [1,1,0,1,0,0,1,0,1,0] => 260 [1,1,0,1,0,0,1,1,0,0] => 390 [1,1,0,1,0,1,0,0,1,0] => 375 [1,1,0,1,0,1,0,1,0,0] => 541 [1,1,0,1,0,1,1,0,0,0] => 843 [1,1,0,1,1,0,0,0,1,0] => 585 [1,1,0,1,1,0,0,1,0,0] => 813 [1,1,0,1,1,0,1,0,0,0] => 1269 [1,1,0,1,1,1,0,0,0,0] => 2121 [1,1,1,0,0,0,1,0,1,0] => 420 [1,1,1,0,0,0,1,1,0,0] => 630 [1,1,1,0,0,1,0,0,1,0] => 585 [1,1,1,0,0,1,0,1,0,0] => 843 [1,1,1,0,0,1,1,0,0,0] => 1269 [1,1,1,0,1,0,0,0,1,0] => 915 [1,1,1,0,1,0,0,1,0,0] => 1269 [1,1,1,0,1,0,1,0,0,0] => 1917 [1,1,1,0,1,1,0,0,0,0] => 3213 [1,1,1,1,0,0,0,0,1,0] => 1575 [1,1,1,1,0,0,0,1,0,0] => 2121 [1,1,1,1,0,0,1,0,0,0] => 3213 [1,1,1,1,0,1,0,0,0,0] => 5397 [1,1,1,1,1,0,0,0,0,0] => 9765 ----------------------------------------------------------------------------- Created: Sep 05, 2018 at 08:45 by Per Alexandersson ----------------------------------------------------------------------------- Last Updated: Sep 25, 2018 at 13:07 by Per Alexandersson