***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew <info@findstat.org> * * * * 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: St001531 ----------------------------------------------------------------------------- Collection: Dyck paths ----------------------------------------------------------------------------- Description: Number of partial orders contained in the poset determined by the Dyck path. A Dyck path determines a poset, where the relations correspond to boxes under the path (seen as a North-East path). This statistic is closely related to unicellular LLT polynomials and their e-expansion. ----------------------------------------------------------------------------- References: [1] Alexandersson, P., Sulzgruber, R. A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions [[arXiv:2004.09198]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1,0,1,0] => 1 [1,1,0,0] => 2 [1,0,1,0,1,0] => 1 [1,0,1,1,0,0] => 2 [1,1,0,0,1,0] => 2 [1,1,0,1,0,0] => 4 [1,1,1,0,0,0] => 7 [1,0,1,0,1,0,1,0] => 1 [1,0,1,0,1,1,0,0] => 2 [1,0,1,1,0,0,1,0] => 2 [1,0,1,1,0,1,0,0] => 4 [1,0,1,1,1,0,0,0] => 7 [1,1,0,0,1,0,1,0] => 2 [1,1,0,0,1,1,0,0] => 4 [1,1,0,1,0,0,1,0] => 4 [1,1,0,1,0,1,0,0] => 8 [1,1,0,1,1,0,0,0] => 14 [1,1,1,0,0,0,1,0] => 7 [1,1,1,0,0,1,0,0] => 14 [1,1,1,0,1,0,0,0] => 25 [1,1,1,1,0,0,0,0] => 40 [1,0,1,0,1,0,1,0,1,0] => 1 [1,0,1,0,1,0,1,1,0,0] => 2 [1,0,1,0,1,1,0,0,1,0] => 2 [1,0,1,0,1,1,0,1,0,0] => 4 [1,0,1,0,1,1,1,0,0,0] => 7 [1,0,1,1,0,0,1,0,1,0] => 2 [1,0,1,1,0,0,1,1,0,0] => 4 [1,0,1,1,0,1,0,0,1,0] => 4 [1,0,1,1,0,1,0,1,0,0] => 8 [1,0,1,1,0,1,1,0,0,0] => 14 [1,0,1,1,1,0,0,0,1,0] => 7 [1,0,1,1,1,0,0,1,0,0] => 14 [1,0,1,1,1,0,1,0,0,0] => 25 [1,0,1,1,1,1,0,0,0,0] => 40 [1,1,0,0,1,0,1,0,1,0] => 2 [1,1,0,0,1,0,1,1,0,0] => 4 [1,1,0,0,1,1,0,0,1,0] => 4 [1,1,0,0,1,1,0,1,0,0] => 8 [1,1,0,0,1,1,1,0,0,0] => 14 [1,1,0,1,0,0,1,0,1,0] => 4 [1,1,0,1,0,0,1,1,0,0] => 8 [1,1,0,1,0,1,0,0,1,0] => 8 [1,1,0,1,0,1,0,1,0,0] => 16 [1,1,0,1,0,1,1,0,0,0] => 28 [1,1,0,1,1,0,0,0,1,0] => 14 [1,1,0,1,1,0,0,1,0,0] => 28 [1,1,0,1,1,0,1,0,0,0] => 50 [1,1,0,1,1,1,0,0,0,0] => 80 [1,1,1,0,0,0,1,0,1,0] => 7 [1,1,1,0,0,0,1,1,0,0] => 14 [1,1,1,0,0,1,0,0,1,0] => 14 [1,1,1,0,0,1,0,1,0,0] => 28 [1,1,1,0,0,1,1,0,0,0] => 49 [1,1,1,0,1,0,0,0,1,0] => 25 [1,1,1,0,1,0,0,1,0,0] => 50 [1,1,1,0,1,0,1,0,0,0] => 89 [1,1,1,0,1,1,0,0,0,0] => 145 [1,1,1,1,0,0,0,0,1,0] => 40 [1,1,1,1,0,0,0,1,0,0] => 80 [1,1,1,1,0,0,1,0,0,0] => 145 [1,1,1,1,0,1,0,0,0,0] => 238 [1,1,1,1,1,0,0,0,0,0] => 357 [1,0,1,0,1,0,1,0,1,0,1,0] => 1 [1,0,1,0,1,0,1,0,1,1,0,0] => 2 [1,0,1,0,1,0,1,1,0,0,1,0] => 2 [1,0,1,0,1,0,1,1,0,1,0,0] => 4 [1,0,1,0,1,0,1,1,1,0,0,0] => 7 [1,0,1,0,1,1,0,0,1,0,1,0] => 2 [1,0,1,0,1,1,0,0,1,1,0,0] => 4 [1,0,1,0,1,1,0,1,0,0,1,0] => 4 [1,0,1,0,1,1,0,1,0,1,0,0] => 8 [1,0,1,0,1,1,0,1,1,0,0,0] => 14 [1,0,1,0,1,1,1,0,0,0,1,0] => 7 [1,0,1,0,1,1,1,0,0,1,0,0] => 14 [1,0,1,0,1,1,1,0,1,0,0,0] => 25 [1,0,1,0,1,1,1,1,0,0,0,0] => 40 [1,0,1,1,0,0,1,0,1,0,1,0] => 2 [1,0,1,1,0,0,1,0,1,1,0,0] => 4 [1,0,1,1,0,0,1,1,0,0,1,0] => 4 [1,0,1,1,0,0,1,1,0,1,0,0] => 8 [1,0,1,1,0,0,1,1,1,0,0,0] => 14 [1,0,1,1,0,1,0,0,1,0,1,0] => 4 [1,0,1,1,0,1,0,0,1,1,0,0] => 8 [1,0,1,1,0,1,0,1,0,0,1,0] => 8 [1,0,1,1,0,1,0,1,0,1,0,0] => 16 [1,0,1,1,0,1,0,1,1,0,0,0] => 28 [1,0,1,1,0,1,1,0,0,0,1,0] => 14 [1,0,1,1,0,1,1,0,0,1,0,0] => 28 [1,0,1,1,0,1,1,0,1,0,0,0] => 50 [1,0,1,1,0,1,1,1,0,0,0,0] => 80 [1,0,1,1,1,0,0,0,1,0,1,0] => 7 [1,0,1,1,1,0,0,0,1,1,0,0] => 14 [1,0,1,1,1,0,0,1,0,0,1,0] => 14 [1,0,1,1,1,0,0,1,0,1,0,0] => 28 [1,0,1,1,1,0,0,1,1,0,0,0] => 49 [1,0,1,1,1,0,1,0,0,0,1,0] => 25 [1,0,1,1,1,0,1,0,0,1,0,0] => 50 [1,0,1,1,1,0,1,0,1,0,0,0] => 89 [1,0,1,1,1,0,1,1,0,0,0,0] => 145 [1,0,1,1,1,1,0,0,0,0,1,0] => 40 [1,0,1,1,1,1,0,0,0,1,0,0] => 80 [1,0,1,1,1,1,0,0,1,0,0,0] => 145 [1,0,1,1,1,1,0,1,0,0,0,0] => 238 [1,0,1,1,1,1,1,0,0,0,0,0] => 357 [1,1,0,0,1,0,1,0,1,0,1,0] => 2 [1,1,0,0,1,0,1,0,1,1,0,0] => 4 [1,1,0,0,1,0,1,1,0,0,1,0] => 4 [1,1,0,0,1,0,1,1,0,1,0,0] => 8 [1,1,0,0,1,0,1,1,1,0,0,0] => 14 [1,1,0,0,1,1,0,0,1,0,1,0] => 4 [1,1,0,0,1,1,0,0,1,1,0,0] => 8 [1,1,0,0,1,1,0,1,0,0,1,0] => 8 [1,1,0,0,1,1,0,1,0,1,0,0] => 16 [1,1,0,0,1,1,0,1,1,0,0,0] => 28 [1,1,0,0,1,1,1,0,0,0,1,0] => 14 [1,1,0,0,1,1,1,0,0,1,0,0] => 28 [1,1,0,0,1,1,1,0,1,0,0,0] => 50 [1,1,0,0,1,1,1,1,0,0,0,0] => 80 [1,1,0,1,0,0,1,0,1,0,1,0] => 4 [1,1,0,1,0,0,1,0,1,1,0,0] => 8 [1,1,0,1,0,0,1,1,0,0,1,0] => 8 [1,1,0,1,0,0,1,1,0,1,0,0] => 16 [1,1,0,1,0,0,1,1,1,0,0,0] => 28 [1,1,0,1,0,1,0,0,1,0,1,0] => 8 [1,1,0,1,0,1,0,0,1,1,0,0] => 16 [1,1,0,1,0,1,0,1,0,0,1,0] => 16 [1,1,0,1,0,1,0,1,0,1,0,0] => 32 [1,1,0,1,0,1,0,1,1,0,0,0] => 56 [1,1,0,1,0,1,1,0,0,0,1,0] => 28 [1,1,0,1,0,1,1,0,0,1,0,0] => 56 [1,1,0,1,0,1,1,0,1,0,0,0] => 100 [1,1,0,1,0,1,1,1,0,0,0,0] => 160 [1,1,0,1,1,0,0,0,1,0,1,0] => 14 [1,1,0,1,1,0,0,0,1,1,0,0] => 28 [1,1,0,1,1,0,0,1,0,0,1,0] => 28 [1,1,0,1,1,0,0,1,0,1,0,0] => 56 [1,1,0,1,1,0,0,1,1,0,0,0] => 98 [1,1,0,1,1,0,1,0,0,0,1,0] => 50 [1,1,0,1,1,0,1,0,0,1,0,0] => 100 [1,1,0,1,1,0,1,0,1,0,0,0] => 178 [1,1,0,1,1,0,1,1,0,0,0,0] => 290 [1,1,0,1,1,1,0,0,0,0,1,0] => 80 [1,1,0,1,1,1,0,0,0,1,0,0] => 160 [1,1,0,1,1,1,0,0,1,0,0,0] => 290 [1,1,0,1,1,1,0,1,0,0,0,0] => 476 [1,1,0,1,1,1,1,0,0,0,0,0] => 714 [1,1,1,0,0,0,1,0,1,0,1,0] => 7 [1,1,1,0,0,0,1,0,1,1,0,0] => 14 [1,1,1,0,0,0,1,1,0,0,1,0] => 14 [1,1,1,0,0,0,1,1,0,1,0,0] => 28 [1,1,1,0,0,0,1,1,1,0,0,0] => 49 [1,1,1,0,0,1,0,0,1,0,1,0] => 14 [1,1,1,0,0,1,0,0,1,1,0,0] => 28 [1,1,1,0,0,1,0,1,0,0,1,0] => 28 [1,1,1,0,0,1,0,1,0,1,0,0] => 56 [1,1,1,0,0,1,0,1,1,0,0,0] => 98 [1,1,1,0,0,1,1,0,0,0,1,0] => 49 [1,1,1,0,0,1,1,0,0,1,0,0] => 98 [1,1,1,0,0,1,1,0,1,0,0,0] => 175 [1,1,1,0,0,1,1,1,0,0,0,0] => 280 [1,1,1,0,1,0,0,0,1,0,1,0] => 25 [1,1,1,0,1,0,0,0,1,1,0,0] => 50 [1,1,1,0,1,0,0,1,0,0,1,0] => 50 [1,1,1,0,1,0,0,1,0,1,0,0] => 100 [1,1,1,0,1,0,0,1,1,0,0,0] => 175 [1,1,1,0,1,0,1,0,0,0,1,0] => 89 [1,1,1,0,1,0,1,0,0,1,0,0] => 178 [1,1,1,0,1,0,1,0,1,0,0,0] => 317 [1,1,1,0,1,0,1,1,0,0,0,0] => 515 [1,1,1,0,1,1,0,0,0,0,1,0] => 145 [1,1,1,0,1,1,0,0,0,1,0,0] => 290 [1,1,1,0,1,1,0,0,1,0,0,0] => 526 [1,1,1,0,1,1,0,1,0,0,0,0] => 859 [1,1,1,0,1,1,1,0,0,0,0,0] => 1309 [1,1,1,1,0,0,0,0,1,0,1,0] => 40 [1,1,1,1,0,0,0,0,1,1,0,0] => 80 [1,1,1,1,0,0,0,1,0,0,1,0] => 80 [1,1,1,1,0,0,0,1,0,1,0,0] => 160 [1,1,1,1,0,0,0,1,1,0,0,0] => 280 [1,1,1,1,0,0,1,0,0,0,1,0] => 145 [1,1,1,1,0,0,1,0,0,1,0,0] => 290 [1,1,1,1,0,0,1,0,1,0,0,0] => 515 [1,1,1,1,0,0,1,1,0,0,0,0] => 850 [1,1,1,1,0,1,0,0,0,0,1,0] => 238 [1,1,1,1,0,1,0,0,0,1,0,0] => 476 [1,1,1,1,0,1,0,0,1,0,0,0] => 859 [1,1,1,1,0,1,0,1,0,0,0,0] => 1427 [1,1,1,1,0,1,1,0,0,0,0,0] => 2194 [1,1,1,1,1,0,0,0,0,0,1,0] => 357 [1,1,1,1,1,0,0,0,0,1,0,0] => 714 [1,1,1,1,1,0,0,0,1,0,0,0] => 1309 [1,1,1,1,1,0,0,1,0,0,0,0] => 2194 [1,1,1,1,1,0,1,0,0,0,0,0] => 3377 [1,1,1,1,1,1,0,0,0,0,0,0] => 4824 ----------------------------------------------------------------------------- Created: Apr 21, 2020 at 11:21 by Per Alexandersson ----------------------------------------------------------------------------- Last Updated: Apr 21, 2020 at 11:21 by Per Alexandersson