*****************************************************************************
*       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: 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