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Statistic identifier: St000166

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Collection: Ordered trees

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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]].

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References: 

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Code:
def statistic(t):
    return t.depth()-1

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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

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Created: Nov 08, 2013 at 21:04 by Viviane Pons

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Last Updated: Feb 17, 2015 at 21:18 by Martin Rubey