***************************************************************************** * 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: St000082 ----------------------------------------------------------------------------- Collection: Binary trees ----------------------------------------------------------------------------- Description: The number of elements smaller than a binary tree in Tamari order. ----------------------------------------------------------------------------- References: [1] Chatel, Grégory, Pons, V. Counting smaller trees in the Tamari order [[MathSciNet:3091011]] ----------------------------------------------------------------------------- Code: def statistic(tree): return len(tree.tamari_smaller()) ----------------------------------------------------------------------------- Statistic values: [.,[.,.]] => 2 [[.,.],.] => 1 [.,[.,[.,.]]] => 5 [.,[[.,.],.]] => 3 [[.,.],[.,.]] => 2 [[.,[.,.]],.] => 2 [[[.,.],.],.] => 1 [.,[.,[.,[.,.]]]] => 14 [.,[.,[[.,.],.]]] => 9 [.,[[.,.],[.,.]]] => 7 [.,[[.,[.,.]],.]] => 7 [.,[[[.,.],.],.]] => 4 [[.,.],[.,[.,.]]] => 5 [[.,.],[[.,.],.]] => 3 [[.,[.,.]],[.,.]] => 4 [[[.,.],.],[.,.]] => 2 [[.,[.,[.,.]]],.] => 5 [[.,[[.,.],.]],.] => 3 [[[.,.],[.,.]],.] => 2 [[[.,[.,.]],.],.] => 2 [[[[.,.],.],.],.] => 1 [.,[.,[.,[.,[.,.]]]]] => 42 [.,[.,[.,[[.,.],.]]]] => 28 [.,[.,[[.,.],[.,.]]]] => 23 [.,[.,[[.,[.,.]],.]]] => 23 [.,[.,[[[.,.],.],.]]] => 14 [.,[[.,.],[.,[.,.]]]] => 19 [.,[[.,.],[[.,.],.]]] => 12 [.,[[.,[.,.]],[.,.]]] => 16 [.,[[[.,.],.],[.,.]]] => 9 [.,[[.,[.,[.,.]]],.]] => 19 [.,[[.,[[.,.],.]],.]] => 12 [.,[[[.,.],[.,.]],.]] => 9 [.,[[[.,[.,.]],.],.]] => 9 [.,[[[[.,.],.],.],.]] => 5 [[.,.],[.,[.,[.,.]]]] => 14 [[.,.],[.,[[.,.],.]]] => 9 [[.,.],[[.,.],[.,.]]] => 7 [[.,.],[[.,[.,.]],.]] => 7 [[.,.],[[[.,.],.],.]] => 4 [[.,[.,.]],[.,[.,.]]] => 10 [[.,[.,.]],[[.,.],.]] => 6 [[[.,.],.],[.,[.,.]]] => 5 [[[.,.],.],[[.,.],.]] => 3 [[.,[.,[.,.]]],[.,.]] => 10 [[.,[[.,.],.]],[.,.]] => 6 [[[.,.],[.,.]],[.,.]] => 4 [[[.,[.,.]],.],[.,.]] => 4 [[[[.,.],.],.],[.,.]] => 2 [[.,[.,[.,[.,.]]]],.] => 14 [[.,[.,[[.,.],.]]],.] => 9 [[.,[[.,.],[.,.]]],.] => 7 [[.,[[.,[.,.]],.]],.] => 7 [[.,[[[.,.],.],.]],.] => 4 [[[.,.],[.,[.,.]]],.] => 5 [[[.,.],[[.,.],.]],.] => 3 [[[.,[.,.]],[.,.]],.] => 4 [[[[.,.],.],[.,.]],.] => 2 [[[.,[.,[.,.]]],.],.] => 5 [[[.,[[.,.],.]],.],.] => 3 [[[[.,.],[.,.]],.],.] => 2 [[[[.,[.,.]],.],.],.] => 2 [[[[[.,.],.],.],.],.] => 1 [.,[.,[.,[.,[.,[.,.]]]]]] => 132 [.,[.,[.,[.,[[.,.],.]]]]] => 90 [.,[.,[.,[[.,.],[.,.]]]]] => 76 [.,[.,[.,[[.,[.,.]],.]]]] => 76 [.,[.,[.,[[[.,.],.],.]]]] => 48 [.,[.,[[.,.],[.,[.,.]]]]] => 66 [.,[.,[[.,.],[[.,.],.]]]] => 43 [.,[.,[[.,[.,.]],[.,.]]]] => 57 [.,[.,[[[.,.],.],[.,.]]]] => 34 [.,[.,[[.,[.,[.,.]]],.]]] => 66 [.,[.,[[.,[[.,.],.]],.]]] => 43 [.,[.,[[[.,.],[.,.]],.]]] => 34 [.,[.,[[[.,[.,.]],.],.]]] => 34 [.,[.,[[[[.,.],.],.],.]]] => 20 [.,[[.,.],[.,[.,[.,.]]]]] => 56 [.,[[.,.],[.,[[.,.],.]]]] => 37 [.,[[.,.],[[.,.],[.,.]]]] => 30 [.,[[.,.],[[.,[.,.]],.]]] => 30 [.,[[.,.],[[[.,.],.],.]]] => 18 [.,[[.,[.,.]],[.,[.,.]]]] => 43 [.,[[.,[.,.]],[[.,.],.]]] => 27 [.,[[[.,.],.],[.,[.,.]]]] => 24 [.,[[[.,.],.],[[.,.],.]]] => 15 [.,[[.,[.,[.,.]]],[.,.]]] => 43 [.,[[.,[[.,.],.]],[.,.]]] => 27 [.,[[[.,.],[.,.]],[.,.]]] => 20 [.,[[[.,[.,.]],.],[.,.]]] => 20 [.,[[[[.,.],.],.],[.,.]]] => 11 [.,[[.,[.,[.,[.,.]]]],.]] => 56 [.,[[.,[.,[[.,.],.]]],.]] => 37 [.,[[.,[[.,.],[.,.]]],.]] => 30 [.,[[.,[[.,[.,.]],.]],.]] => 30 [.,[[.,[[[.,.],.],.]],.]] => 18 [.,[[[.,.],[.,[.,.]]],.]] => 24 [.,[[[.,.],[[.,.],.]],.]] => 15 [.,[[[.,[.,.]],[.,.]],.]] => 20 [.,[[[[.,.],.],[.,.]],.]] => 11 [.,[[[.,[.,[.,.]]],.],.]] => 24 [.,[[[.,[[.,.],.]],.],.]] => 15 [.,[[[[.,.],[.,.]],.],.]] => 11 [.,[[[[.,[.,.]],.],.],.]] => 11 [.,[[[[[.,.],.],.],.],.]] => 6 [[.,.],[.,[.,[.,[.,.]]]]] => 42 [[.,.],[.,[.,[[.,.],.]]]] => 28 [[.,.],[.,[[.,.],[.,.]]]] => 23 [[.,.],[.,[[.,[.,.]],.]]] => 23 [[.,.],[.,[[[.,.],.],.]]] => 14 [[.,.],[[.,.],[.,[.,.]]]] => 19 [[.,.],[[.,.],[[.,.],.]]] => 12 [[.,.],[[.,[.,.]],[.,.]]] => 16 [[.,.],[[[.,.],.],[.,.]]] => 9 [[.,.],[[.,[.,[.,.]]],.]] => 19 [[.,.],[[.,[[.,.],.]],.]] => 12 [[.,.],[[[.,.],[.,.]],.]] => 9 [[.,.],[[[.,[.,.]],.],.]] => 9 [[.,.],[[[[.,.],.],.],.]] => 5 [[.,[.,.]],[.,[.,[.,.]]]] => 28 [[.,[.,.]],[.,[[.,.],.]]] => 18 [[.,[.,.]],[[.,.],[.,.]]] => 14 [[.,[.,.]],[[.,[.,.]],.]] => 14 [[.,[.,.]],[[[.,.],.],.]] => 8 [[[.,.],.],[.,[.,[.,.]]]] => 14 [[[.,.],.],[.,[[.,.],.]]] => 9 [[[.,.],.],[[.,.],[.,.]]] => 7 [[[.,.],.],[[.,[.,.]],.]] => 7 [[[.,.],.],[[[.,.],.],.]] => 4 [[.,[.,[.,.]]],[.,[.,.]]] => 25 [[.,[.,[.,.]]],[[.,.],.]] => 15 [[.,[[.,.],.]],[.,[.,.]]] => 15 [[.,[[.,.],.]],[[.,.],.]] => 9 [[[.,.],[.,.]],[.,[.,.]]] => 10 [[[.,.],[.,.]],[[.,.],.]] => 6 [[[.,[.,.]],.],[.,[.,.]]] => 10 [[[.,[.,.]],.],[[.,.],.]] => 6 [[[[.,.],.],.],[.,[.,.]]] => 5 [[[[.,.],.],.],[[.,.],.]] => 3 [[.,[.,[.,[.,.]]]],[.,.]] => 28 [[.,[.,[[.,.],.]]],[.,.]] => 18 [[.,[[.,.],[.,.]]],[.,.]] => 14 [[.,[[.,[.,.]],.]],[.,.]] => 14 [[.,[[[.,.],.],.]],[.,.]] => 8 [[[.,.],[.,[.,.]]],[.,.]] => 10 [[[.,.],[[.,.],.]],[.,.]] => 6 [[[.,[.,.]],[.,.]],[.,.]] => 8 [[[[.,.],.],[.,.]],[.,.]] => 4 [[[.,[.,[.,.]]],.],[.,.]] => 10 [[[.,[[.,.],.]],.],[.,.]] => 6 [[[[.,.],[.,.]],.],[.,.]] => 4 [[[[.,[.,.]],.],.],[.,.]] => 4 [[[[[.,.],.],.],.],[.,.]] => 2 [[.,[.,[.,[.,[.,.]]]]],.] => 42 [[.,[.,[.,[[.,.],.]]]],.] => 28 [[.,[.,[[.,.],[.,.]]]],.] => 23 [[.,[.,[[.,[.,.]],.]]],.] => 23 [[.,[.,[[[.,.],.],.]]],.] => 14 [[.,[[.,.],[.,[.,.]]]],.] => 19 [[.,[[.,.],[[.,.],.]]],.] => 12 [[.,[[.,[.,.]],[.,.]]],.] => 16 [[.,[[[.,.],.],[.,.]]],.] => 9 [[.,[[.,[.,[.,.]]],.]],.] => 19 [[.,[[.,[[.,.],.]],.]],.] => 12 [[.,[[[.,.],[.,.]],.]],.] => 9 [[.,[[[.,[.,.]],.],.]],.] => 9 [[.,[[[[.,.],.],.],.]],.] => 5 [[[.,.],[.,[.,[.,.]]]],.] => 14 [[[.,.],[.,[[.,.],.]]],.] => 9 [[[.,.],[[.,.],[.,.]]],.] => 7 [[[.,.],[[.,[.,.]],.]],.] => 7 [[[.,.],[[[.,.],.],.]],.] => 4 [[[.,[.,.]],[.,[.,.]]],.] => 10 [[[.,[.,.]],[[.,.],.]],.] => 6 [[[[.,.],.],[.,[.,.]]],.] => 5 [[[[.,.],.],[[.,.],.]],.] => 3 [[[.,[.,[.,.]]],[.,.]],.] => 10 [[[.,[[.,.],.]],[.,.]],.] => 6 [[[[.,.],[.,.]],[.,.]],.] => 4 [[[[.,[.,.]],.],[.,.]],.] => 4 [[[[[.,.],.],.],[.,.]],.] => 2 [[[.,[.,[.,[.,.]]]],.],.] => 14 [[[.,[.,[[.,.],.]]],.],.] => 9 [[[.,[[.,.],[.,.]]],.],.] => 7 [[[.,[[.,[.,.]],.]],.],.] => 7 [[[.,[[[.,.],.],.]],.],.] => 4 [[[[.,.],[.,[.,.]]],.],.] => 5 [[[[.,.],[[.,.],.]],.],.] => 3 [[[[.,[.,.]],[.,.]],.],.] => 4 [[[[[.,.],.],[.,.]],.],.] => 2 [[[[.,[.,[.,.]]],.],.],.] => 5 [[[[.,[[.,.],.]],.],.],.] => 3 [[[[[.,.],[.,.]],.],.],.] => 2 [[[[[.,[.,.]],.],.],.],.] => 2 [[[[[[.,.],.],.],.],.],.] => 1 ----------------------------------------------------------------------------- Created: Jun 13, 2013 at 09:55 by Viviane Pons ----------------------------------------------------------------------------- Last Updated: Oct 17, 2015 at 10:48 by Christian Stump