***************************************************************************** * 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: St000166 ----------------------------------------------------------------------------- Collection: Ordered trees ----------------------------------------------------------------------------- Description: The depth minus 1 of an ordered tree. The ordered trees of size $n$ are bijection with the Dyck paths of size $n-1$, and this statistic then corresponds to [[St000013]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(t): return t.depth()-1 ----------------------------------------------------------------------------- Statistic values: [[]] => 1 [[],[]] => 1 [[[]]] => 2 [[],[],[]] => 1 [[],[[]]] => 2 [[[]],[]] => 2 [[[],[]]] => 2 [[[[]]]] => 3 [[],[],[],[]] => 1 [[],[],[[]]] => 2 [[],[[]],[]] => 2 [[],[[],[]]] => 2 [[],[[[]]]] => 3 [[[]],[],[]] => 2 [[[]],[[]]] => 2 [[[],[]],[]] => 2 [[[[]]],[]] => 3 [[[],[],[]]] => 2 [[[],[[]]]] => 3 [[[[]],[]]] => 3 [[[[],[]]]] => 3 [[[[[]]]]] => 4 [[],[],[],[],[]] => 1 [[],[],[],[[]]] => 2 [[],[],[[]],[]] => 2 [[],[],[[],[]]] => 2 [[],[],[[[]]]] => 3 [[],[[]],[],[]] => 2 [[],[[]],[[]]] => 2 [[],[[],[]],[]] => 2 [[],[[[]]],[]] => 3 [[],[[],[],[]]] => 2 [[],[[],[[]]]] => 3 [[],[[[]],[]]] => 3 [[],[[[],[]]]] => 3 [[],[[[[]]]]] => 4 [[[]],[],[],[]] => 2 [[[]],[],[[]]] => 2 [[[]],[[]],[]] => 2 [[[]],[[],[]]] => 2 [[[]],[[[]]]] => 3 [[[],[]],[],[]] => 2 [[[[]]],[],[]] => 3 [[[],[]],[[]]] => 2 [[[[]]],[[]]] => 3 [[[],[],[]],[]] => 2 [[[],[[]]],[]] => 3 [[[[]],[]],[]] => 3 [[[[],[]]],[]] => 3 [[[[[]]]],[]] => 4 [[[],[],[],[]]] => 2 [[[],[],[[]]]] => 3 [[[],[[]],[]]] => 3 [[[],[[],[]]]] => 3 [[[],[[[]]]]] => 4 [[[[]],[],[]]] => 3 [[[[]],[[]]]] => 3 [[[[],[]],[]]] => 3 [[[[[]]],[]]] => 4 [[[[],[],[]]]] => 3 [[[[],[[]]]]] => 4 [[[[[]],[]]]] => 4 [[[[[],[]]]]] => 4 [[[[[[]]]]]] => 5 [[],[],[],[],[],[]] => 1 [[],[],[],[],[[]]] => 2 [[],[],[],[[]],[]] => 2 [[],[],[],[[],[]]] => 2 [[],[],[],[[[]]]] => 3 [[],[],[[]],[],[]] => 2 [[],[],[[]],[[]]] => 2 [[],[],[[],[]],[]] => 2 [[],[],[[[]]],[]] => 3 [[],[],[[],[],[]]] => 2 [[],[],[[],[[]]]] => 3 [[],[],[[[]],[]]] => 3 [[],[],[[[],[]]]] => 3 [[],[],[[[[]]]]] => 4 [[],[[]],[],[],[]] => 2 [[],[[]],[],[[]]] => 2 [[],[[]],[[]],[]] => 2 [[],[[]],[[],[]]] => 2 [[],[[]],[[[]]]] => 3 [[],[[],[]],[],[]] => 2 [[],[[[]]],[],[]] => 3 [[],[[],[]],[[]]] => 2 [[],[[[]]],[[]]] => 3 [[],[[],[],[]],[]] => 2 [[],[[],[[]]],[]] => 3 [[],[[[]],[]],[]] => 3 [[],[[[],[]]],[]] => 3 [[],[[[[]]]],[]] => 4 [[],[[],[],[],[]]] => 2 [[],[[],[],[[]]]] => 3 [[],[[],[[]],[]]] => 3 [[],[[],[[],[]]]] => 3 [[],[[],[[[]]]]] => 4 [[],[[[]],[],[]]] => 3 [[],[[[]],[[]]]] => 3 [[],[[[],[]],[]]] => 3 [[],[[[[]]],[]]] => 4 [[],[[[],[],[]]]] => 3 [[],[[[],[[]]]]] => 4 [[],[[[[]],[]]]] => 4 [[],[[[[],[]]]]] => 4 [[],[[[[[]]]]]] => 5 [[[]],[],[],[],[]] => 2 [[[]],[],[],[[]]] => 2 [[[]],[],[[]],[]] => 2 [[[]],[],[[],[]]] => 2 [[[]],[],[[[]]]] => 3 [[[]],[[]],[],[]] => 2 [[[]],[[]],[[]]] => 2 [[[]],[[],[]],[]] => 2 [[[]],[[[]]],[]] => 3 [[[]],[[],[],[]]] => 2 [[[]],[[],[[]]]] => 3 [[[]],[[[]],[]]] => 3 [[[]],[[[],[]]]] => 3 [[[]],[[[[]]]]] => 4 [[[],[]],[],[],[]] => 2 [[[[]]],[],[],[]] => 3 [[[],[]],[],[[]]] => 2 [[[[]]],[],[[]]] => 3 [[[],[]],[[]],[]] => 2 [[[[]]],[[]],[]] => 3 [[[],[]],[[],[]]] => 2 [[[],[]],[[[]]]] => 3 [[[[]]],[[],[]]] => 3 [[[[]]],[[[]]]] => 3 [[[],[],[]],[],[]] => 2 [[[],[[]]],[],[]] => 3 [[[[]],[]],[],[]] => 3 [[[[],[]]],[],[]] => 3 [[[[[]]]],[],[]] => 4 [[[],[],[]],[[]]] => 2 [[[],[[]]],[[]]] => 3 [[[[]],[]],[[]]] => 3 [[[[],[]]],[[]]] => 3 [[[[[]]]],[[]]] => 4 [[[],[],[],[]],[]] => 2 [[[],[],[[]]],[]] => 3 [[[],[[]],[]],[]] => 3 [[[],[[],[]]],[]] => 3 [[[],[[[]]]],[]] => 4 [[[[]],[],[]],[]] => 3 [[[[]],[[]]],[]] => 3 [[[[],[]],[]],[]] => 3 [[[[[]]],[]],[]] => 4 [[[[],[],[]]],[]] => 3 [[[[],[[]]]],[]] => 4 [[[[[]],[]]],[]] => 4 [[[[[],[]]]],[]] => 4 [[[[[[]]]]],[]] => 5 [[[],[],[],[],[]]] => 2 [[[],[],[],[[]]]] => 3 [[[],[],[[]],[]]] => 3 [[[],[],[[],[]]]] => 3 [[[],[],[[[]]]]] => 4 [[[],[[]],[],[]]] => 3 [[[],[[]],[[]]]] => 3 [[[],[[],[]],[]]] => 3 [[[],[[[]]],[]]] => 4 [[[],[[],[],[]]]] => 3 [[[],[[],[[]]]]] => 4 [[[],[[[]],[]]]] => 4 [[[],[[[],[]]]]] => 4 [[[],[[[[]]]]]] => 5 [[[[]],[],[],[]]] => 3 [[[[]],[],[[]]]] => 3 [[[[]],[[]],[]]] => 3 [[[[]],[[],[]]]] => 3 [[[[]],[[[]]]]] => 4 [[[[],[]],[],[]]] => 3 [[[[[]]],[],[]]] => 4 [[[[],[]],[[]]]] => 3 [[[[[]]],[[]]]] => 4 [[[[],[],[]],[]]] => 3 [[[[],[[]]],[]]] => 4 [[[[[]],[]],[]]] => 4 [[[[[],[]]],[]]] => 4 [[[[[[]]]],[]]] => 5 [[[[],[],[],[]]]] => 3 [[[[],[],[[]]]]] => 4 [[[[],[[]],[]]]] => 4 [[[[],[[],[]]]]] => 4 [[[[],[[[]]]]]] => 5 [[[[[]],[],[]]]] => 4 [[[[[]],[[]]]]] => 4 [[[[[],[]],[]]]] => 4 [[[[[[]]],[]]]] => 5 [[[[[],[],[]]]]] => 4 [[[[[],[[]]]]]] => 5 [[[[[[]],[]]]]] => 5 [[[[[[],[]]]]]] => 5 [[[[[[[]]]]]]] => 6 ----------------------------------------------------------------------------- Created: Nov 08, 2013 at 21:04 by Viviane Pons ----------------------------------------------------------------------------- Last Updated: Feb 17, 2015 at 21:18 by Martin Rubey