***************************************************************************** * 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: St001243 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. In other words, 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. Consider the sum $$S_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1).$$ This statistic is $S_\Gamma$. It is still an open problem to find a combinatorial description of the above Schur expansion, a first step would be to find a family of combinatorial objects to sum over. ----------------------------------------------------------------------------- References: [1] Alexandersson, P., Panova, G. LLT polynomials, chromatic quasisymmetric functions and graphs with cycles [[arXiv:1705.10353]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1,0] => 1 [1,0,1,0] => 2 [1,1,0,0] => 3 [1,0,1,0,1,0] => 4 [1,0,1,1,0,0] => 6 [1,1,0,0,1,0] => 6 [1,1,0,1,0,0] => 9 [1,1,1,0,0,0] => 15 [1,0,1,0,1,0,1,0] => 10 [1,0,1,0,1,1,0,0] => 15 [1,0,1,1,0,0,1,0] => 15 [1,0,1,1,0,1,0,0] => 22 [1,0,1,1,1,0,0,0] => 36 [1,1,0,0,1,0,1,0] => 15 [1,1,0,0,1,1,0,0] => 23 [1,1,0,1,0,0,1,0] => 22 [1,1,0,1,0,1,0,0] => 33 [1,1,0,1,1,0,0,0] => 53 [1,1,1,0,0,0,1,0] => 36 [1,1,1,0,0,1,0,0] => 53 [1,1,1,0,1,0,0,0] => 87 [1,1,1,1,0,0,0,0] => 155 [1,0,1,0,1,0,1,0,1,0] => 26 [1,0,1,0,1,0,1,1,0,0] => 39 [1,0,1,0,1,1,0,0,1,0] => 39 [1,0,1,0,1,1,0,1,0,0] => 57 [1,0,1,0,1,1,1,0,0,0] => 93 [1,0,1,1,0,0,1,0,1,0] => 39 [1,0,1,1,0,0,1,1,0,0] => 59 [1,0,1,1,0,1,0,0,1,0] => 57 [1,0,1,1,0,1,0,1,0,0] => 84 [1,0,1,1,0,1,1,0,0,0] => 134 [1,0,1,1,1,0,0,0,1,0] => 93 [1,0,1,1,1,0,0,1,0,0] => 134 [1,0,1,1,1,0,1,0,0,0] => 216 [1,0,1,1,1,1,0,0,0,0] => 380 [1,1,0,0,1,0,1,0,1,0] => 39 [1,1,0,0,1,0,1,1,0,0] => 59 [1,1,0,0,1,1,0,0,1,0] => 59 [1,1,0,0,1,1,0,1,0,0] => 87 [1,1,0,0,1,1,1,0,0,0] => 143 [1,1,0,1,0,0,1,0,1,0] => 57 [1,1,0,1,0,0,1,1,0,0] => 87 [1,1,0,1,0,1,0,0,1,0] => 84 [1,1,0,1,0,1,0,1,0,0] => 125 [1,1,0,1,0,1,1,0,0,0] => 201 [1,1,0,1,1,0,0,0,1,0] => 134 [1,1,0,1,1,0,0,1,0,0] => 195 [1,1,0,1,1,0,1,0,0,0] => 317 [1,1,0,1,1,1,0,0,0,0] => 549 [1,1,1,0,0,0,1,0,1,0] => 93 [1,1,1,0,0,0,1,1,0,0] => 143 [1,1,1,0,0,1,0,0,1,0] => 134 [1,1,1,0,0,1,0,1,0,0] => 201 [1,1,1,0,0,1,1,0,0,0] => 317 [1,1,1,0,1,0,0,0,1,0] => 216 [1,1,1,0,1,0,0,1,0,0] => 317 [1,1,1,0,1,0,1,0,0,0] => 507 [1,1,1,0,1,1,0,0,0,0] => 887 [1,1,1,1,0,0,0,0,1,0] => 380 [1,1,1,1,0,0,0,1,0,0] => 549 [1,1,1,1,0,0,1,0,0,0] => 887 [1,1,1,1,0,1,0,0,0,0] => 1563 [1,1,1,1,1,0,0,0,0,0] => 2915 ----------------------------------------------------------------------------- Created: Sep 05, 2018 at 08:58 by Per Alexandersson ----------------------------------------------------------------------------- Last Updated: Sep 05, 2018 at 08:58 by Per Alexandersson