***************************************************************************** * 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: St000252 ----------------------------------------------------------------------------- Collection: Binary trees ----------------------------------------------------------------------------- Description: The number of nodes of degree 3 of a binary tree. Equivalently, the number of internal triangles in the associated triangulation of an $(n+2)$-gon. ----------------------------------------------------------------------------- References: [1] Dochtermann, A. Face rings of cycles, associahedra, and standard Young tableaux [[arXiv:1503.06243]] ----------------------------------------------------------------------------- Code: def statistic(T): return sum(statistic_aux(C) for C in T) def statistic_aux(T): if T == BinaryTree() or (T[0] == BinaryTree() and T[1] == BinaryTree()): return 0 if T[0] == BinaryTree(): return statistic_aux(T[1]) if T[1] == BinaryTree(): return statistic_aux(T[0]) return 1 + statistic_aux(T[0]) + statistic_aux(T[1]) ----------------------------------------------------------------------------- Statistic values: [.,.] => 0 [.,[.,.]] => 0 [[.,.],.] => 0 [.,[.,[.,.]]] => 0 [.,[[.,.],.]] => 0 [[.,.],[.,.]] => 0 [[.,[.,.]],.] => 0 [[[.,.],.],.] => 0 [.,[.,[.,[.,.]]]] => 0 [.,[.,[[.,.],.]]] => 0 [.,[[.,.],[.,.]]] => 1 [.,[[.,[.,.]],.]] => 0 [.,[[[.,.],.],.]] => 0 [[.,.],[.,[.,.]]] => 0 [[.,.],[[.,.],.]] => 0 [[.,[.,.]],[.,.]] => 0 [[[.,.],.],[.,.]] => 0 [[.,[.,[.,.]]],.] => 0 [[.,[[.,.],.]],.] => 0 [[[.,.],[.,.]],.] => 1 [[[.,[.,.]],.],.] => 0 [[[[.,.],.],.],.] => 0 [.,[.,[.,[.,[.,.]]]]] => 0 [.,[.,[.,[[.,.],.]]]] => 0 [.,[.,[[.,.],[.,.]]]] => 1 [.,[.,[[.,[.,.]],.]]] => 0 [.,[.,[[[.,.],.],.]]] => 0 [.,[[.,.],[.,[.,.]]]] => 1 [.,[[.,.],[[.,.],.]]] => 1 [.,[[.,[.,.]],[.,.]]] => 1 [.,[[[.,.],.],[.,.]]] => 1 [.,[[.,[.,[.,.]]],.]] => 0 [.,[[.,[[.,.],.]],.]] => 0 [.,[[[.,.],[.,.]],.]] => 1 [.,[[[.,[.,.]],.],.]] => 0 [.,[[[[.,.],.],.],.]] => 0 [[.,.],[.,[.,[.,.]]]] => 0 [[.,.],[.,[[.,.],.]]] => 0 [[.,.],[[.,.],[.,.]]] => 1 [[.,.],[[.,[.,.]],.]] => 0 [[.,.],[[[.,.],.],.]] => 0 [[.,[.,.]],[.,[.,.]]] => 0 [[.,[.,.]],[[.,.],.]] => 0 [[[.,.],.],[.,[.,.]]] => 0 [[[.,.],.],[[.,.],.]] => 0 [[.,[.,[.,.]]],[.,.]] => 0 [[.,[[.,.],.]],[.,.]] => 0 [[[.,.],[.,.]],[.,.]] => 1 [[[.,[.,.]],.],[.,.]] => 0 [[[[.,.],.],.],[.,.]] => 0 [[.,[.,[.,[.,.]]]],.] => 0 [[.,[.,[[.,.],.]]],.] => 0 [[.,[[.,.],[.,.]]],.] => 1 [[.,[[.,[.,.]],.]],.] => 0 [[.,[[[.,.],.],.]],.] => 0 [[[.,.],[.,[.,.]]],.] => 1 [[[.,.],[[.,.],.]],.] => 1 [[[.,[.,.]],[.,.]],.] => 1 [[[[.,.],.],[.,.]],.] => 1 [[[.,[.,[.,.]]],.],.] => 0 [[[.,[[.,.],.]],.],.] => 0 [[[[.,.],[.,.]],.],.] => 1 [[[[.,[.,.]],.],.],.] => 0 [[[[[.,.],.],.],.],.] => 0 [.,[.,[.,[.,[.,[.,.]]]]]] => 0 [.,[.,[.,[.,[[.,.],.]]]]] => 0 [.,[.,[.,[[.,.],[.,.]]]]] => 1 [.,[.,[.,[[.,[.,.]],.]]]] => 0 [.,[.,[.,[[[.,.],.],.]]]] => 0 [.,[.,[[.,.],[.,[.,.]]]]] => 1 [.,[.,[[.,.],[[.,.],.]]]] => 1 [.,[.,[[.,[.,.]],[.,.]]]] => 1 [.,[.,[[[.,.],.],[.,.]]]] => 1 [.,[.,[[.,[.,[.,.]]],.]]] => 0 [.,[.,[[.,[[.,.],.]],.]]] => 0 [.,[.,[[[.,.],[.,.]],.]]] => 1 [.,[.,[[[.,[.,.]],.],.]]] => 0 [.,[.,[[[[.,.],.],.],.]]] => 0 [.,[[.,.],[.,[.,[.,.]]]]] => 1 [.,[[.,.],[.,[[.,.],.]]]] => 1 [.,[[.,.],[[.,.],[.,.]]]] => 2 [.,[[.,.],[[.,[.,.]],.]]] => 1 [.,[[.,.],[[[.,.],.],.]]] => 1 [.,[[.,[.,.]],[.,[.,.]]]] => 1 [.,[[.,[.,.]],[[.,.],.]]] => 1 [.,[[[.,.],.],[.,[.,.]]]] => 1 [.,[[[.,.],.],[[.,.],.]]] => 1 [.,[[.,[.,[.,.]]],[.,.]]] => 1 [.,[[.,[[.,.],.]],[.,.]]] => 1 [.,[[[.,.],[.,.]],[.,.]]] => 2 [.,[[[.,[.,.]],.],[.,.]]] => 1 [.,[[[[.,.],.],.],[.,.]]] => 1 [.,[[.,[.,[.,[.,.]]]],.]] => 0 [.,[[.,[.,[[.,.],.]]],.]] => 0 [.,[[.,[[.,.],[.,.]]],.]] => 1 [.,[[.,[[.,[.,.]],.]],.]] => 0 [.,[[.,[[[.,.],.],.]],.]] => 0 [.,[[[.,.],[.,[.,.]]],.]] => 1 [.,[[[.,.],[[.,.],.]],.]] => 1 [.,[[[.,[.,.]],[.,.]],.]] => 1 [.,[[[[.,.],.],[.,.]],.]] => 1 [.,[[[.,[.,[.,.]]],.],.]] => 0 [.,[[[.,[[.,.],.]],.],.]] => 0 [.,[[[[.,.],[.,.]],.],.]] => 1 [.,[[[[.,[.,.]],.],.],.]] => 0 [.,[[[[[.,.],.],.],.],.]] => 0 [[.,.],[.,[.,[.,[.,.]]]]] => 0 [[.,.],[.,[.,[[.,.],.]]]] => 0 [[.,.],[.,[[.,.],[.,.]]]] => 1 [[.,.],[.,[[.,[.,.]],.]]] => 0 [[.,.],[.,[[[.,.],.],.]]] => 0 [[.,.],[[.,.],[.,[.,.]]]] => 1 [[.,.],[[.,.],[[.,.],.]]] => 1 [[.,.],[[.,[.,.]],[.,.]]] => 1 [[.,.],[[[.,.],.],[.,.]]] => 1 [[.,.],[[.,[.,[.,.]]],.]] => 0 [[.,.],[[.,[[.,.],.]],.]] => 0 [[.,.],[[[.,.],[.,.]],.]] => 1 [[.,.],[[[.,[.,.]],.],.]] => 0 [[.,.],[[[[.,.],.],.],.]] => 0 [[.,[.,.]],[.,[.,[.,.]]]] => 0 [[.,[.,.]],[.,[[.,.],.]]] => 0 [[.,[.,.]],[[.,.],[.,.]]] => 1 [[.,[.,.]],[[.,[.,.]],.]] => 0 [[.,[.,.]],[[[.,.],.],.]] => 0 [[[.,.],.],[.,[.,[.,.]]]] => 0 [[[.,.],.],[.,[[.,.],.]]] => 0 [[[.,.],.],[[.,.],[.,.]]] => 1 [[[.,.],.],[[.,[.,.]],.]] => 0 [[[.,.],.],[[[.,.],.],.]] => 0 [[.,[.,[.,.]]],[.,[.,.]]] => 0 [[.,[.,[.,.]]],[[.,.],.]] => 0 [[.,[[.,.],.]],[.,[.,.]]] => 0 [[.,[[.,.],.]],[[.,.],.]] => 0 [[[.,.],[.,.]],[.,[.,.]]] => 1 [[[.,.],[.,.]],[[.,.],.]] => 1 [[[.,[.,.]],.],[.,[.,.]]] => 0 [[[.,[.,.]],.],[[.,.],.]] => 0 [[[[.,.],.],.],[.,[.,.]]] => 0 [[[[.,.],.],.],[[.,.],.]] => 0 [[.,[.,[.,[.,.]]]],[.,.]] => 0 [[.,[.,[[.,.],.]]],[.,.]] => 0 [[.,[[.,.],[.,.]]],[.,.]] => 1 [[.,[[.,[.,.]],.]],[.,.]] => 0 [[.,[[[.,.],.],.]],[.,.]] => 0 [[[.,.],[.,[.,.]]],[.,.]] => 1 [[[.,.],[[.,.],.]],[.,.]] => 1 [[[.,[.,.]],[.,.]],[.,.]] => 1 [[[[.,.],.],[.,.]],[.,.]] => 1 [[[.,[.,[.,.]]],.],[.,.]] => 0 [[[.,[[.,.],.]],.],[.,.]] => 0 [[[[.,.],[.,.]],.],[.,.]] => 1 [[[[.,[.,.]],.],.],[.,.]] => 0 [[[[[.,.],.],.],.],[.,.]] => 0 [[.,[.,[.,[.,[.,.]]]]],.] => 0 [[.,[.,[.,[[.,.],.]]]],.] => 0 [[.,[.,[[.,.],[.,.]]]],.] => 1 [[.,[.,[[.,[.,.]],.]]],.] => 0 [[.,[.,[[[.,.],.],.]]],.] => 0 [[.,[[.,.],[.,[.,.]]]],.] => 1 [[.,[[.,.],[[.,.],.]]],.] => 1 [[.,[[.,[.,.]],[.,.]]],.] => 1 [[.,[[[.,.],.],[.,.]]],.] => 1 [[.,[[.,[.,[.,.]]],.]],.] => 0 [[.,[[.,[[.,.],.]],.]],.] => 0 [[.,[[[.,.],[.,.]],.]],.] => 1 [[.,[[[.,[.,.]],.],.]],.] => 0 [[.,[[[[.,.],.],.],.]],.] => 0 [[[.,.],[.,[.,[.,.]]]],.] => 1 [[[.,.],[.,[[.,.],.]]],.] => 1 [[[.,.],[[.,.],[.,.]]],.] => 2 [[[.,.],[[.,[.,.]],.]],.] => 1 [[[.,.],[[[.,.],.],.]],.] => 1 [[[.,[.,.]],[.,[.,.]]],.] => 1 [[[.,[.,.]],[[.,.],.]],.] => 1 [[[[.,.],.],[.,[.,.]]],.] => 1 [[[[.,.],.],[[.,.],.]],.] => 1 [[[.,[.,[.,.]]],[.,.]],.] => 1 [[[.,[[.,.],.]],[.,.]],.] => 1 [[[[.,.],[.,.]],[.,.]],.] => 2 [[[[.,[.,.]],.],[.,.]],.] => 1 [[[[[.,.],.],.],[.,.]],.] => 1 [[[.,[.,[.,[.,.]]]],.],.] => 0 [[[.,[.,[[.,.],.]]],.],.] => 0 [[[.,[[.,.],[.,.]]],.],.] => 1 [[[.,[[.,[.,.]],.]],.],.] => 0 [[[.,[[[.,.],.],.]],.],.] => 0 [[[[.,.],[.,[.,.]]],.],.] => 1 [[[[.,.],[[.,.],.]],.],.] => 1 [[[[.,[.,.]],[.,.]],.],.] => 1 [[[[[.,.],.],[.,.]],.],.] => 1 [[[[.,[.,[.,.]]],.],.],.] => 0 [[[[.,[[.,.],.]],.],.],.] => 0 [[[[[.,.],[.,.]],.],.],.] => 1 [[[[[.,[.,.]],.],.],.],.] => 0 [[[[[[.,.],.],.],.],.],.] => 0 ----------------------------------------------------------------------------- Created: May 27, 2015 at 15:00 by Anton Dochtermann ----------------------------------------------------------------------------- Last Updated: Aug 05, 2016 at 20:11 by Martin Rubey