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Identifier

St000975: Ordered trees ⟶ ℤ

Values

[[]] => 1

[[],[]] => 2

[[[]]] => 2

[[],[],[]] => 2

[[],[[]]] => 3

[[[]],[]] => 3

[[[],[]]] => 3

[[[[]]]] => 3

[[],[],[],[]] => 2

[[],[],[[]]] => 3

[[],[[]],[]] => 2

[[],[[],[]]] => 3

[[],[[[]]]] => 4

[[[]],[],[]] => 3

[[[]],[[]]] => 4

[[[],[]],[]] => 3

[[[[]]],[]] => 4

[[[],[],[]]] => 3

[[[],[[]]]] => 4

[[[[]],[]]] => 4

[[[[],[]]]] => 4

[[[[[]]]]] => 4

[[],[],[],[],[]] => 2

[[],[],[],[[]]] => 3

[[],[],[[]],[]] => 2

[[],[],[[],[]]] => 3

[[],[],[[[]]]] => 4

[[],[[]],[],[]] => 2

[[],[[]],[[]]] => 3

[[],[[],[]],[]] => 2

[[],[[[]]],[]] => 2

[[],[[],[],[]]] => 3

[[],[[],[[]]]] => 4

[[],[[[]],[]]] => 3

[[],[[[],[]]]] => 4

[[],[[[[]]]]] => 5

[[[]],[],[],[]] => 3

[[[]],[],[[]]] => 4

[[[]],[[]],[]] => 3

[[[]],[[],[]]] => 4

[[[]],[[[]]]] => 5

[[[],[]],[],[]] => 3

[[[[]]],[],[]] => 4

[[[],[]],[[]]] => 4

[[[[]]],[[]]] => 5

[[[],[],[]],[]] => 3

[[[],[[]]],[]] => 3

[[[[]],[]],[]] => 4

[[[[],[]]],[]] => 4

[[[[[]]]],[]] => 5

[[[],[],[],[]]] => 3

[[[],[],[[]]]] => 4

[[[],[[]],[]]] => 3

[[[],[[],[]]]] => 4

[[[],[[[]]]]] => 5

[[[[]],[],[]]] => 4

[[[[]],[[]]]] => 5

[[[[],[]],[]]] => 4

[[[[[]]],[]]] => 5

[[[[],[],[]]]] => 4

[[[[],[[]]]]] => 5

[[[[[]],[]]]] => 5

[[[[[],[]]]]] => 5

[[[[[[]]]]]] => 5

[[],[],[],[],[],[]] => 2

[[],[],[],[],[[]]] => 3

[[],[],[],[[]],[]] => 2

[[],[],[],[[],[]]] => 3

[[],[],[],[[[]]]] => 4

[[],[],[[]],[],[]] => 2

[[],[],[[]],[[]]] => 3

[[],[],[[],[]],[]] => 2

[[],[],[[[]]],[]] => 2

[[],[],[[],[],[]]] => 3

[[],[],[[],[[]]]] => 4

[[],[],[[[]],[]]] => 3

[[],[],[[[],[]]]] => 4

[[],[],[[[[]]]]] => 5

[[],[[]],[],[],[]] => 2

[[],[[]],[],[[]]] => 3

[[],[[]],[[]],[]] => 2

[[],[[]],[[],[]]] => 3

[[],[[]],[[[]]]] => 4

[[],[[],[]],[],[]] => 2

[[],[[[]]],[],[]] => 2

[[],[[],[]],[[]]] => 3

[[],[[[]]],[[]]] => 3

[[],[[],[],[]],[]] => 2

[[],[[],[[]]],[]] => 2

[[],[[[]],[]],[]] => 2

[[],[[[],[]]],[]] => 2

[[],[[[[]]]],[]] => 2

[[],[[],[],[],[]]] => 3

[[],[[],[],[[]]]] => 4

[[],[[],[[]],[]]] => 3

[[],[[],[[],[]]]] => 4

[[],[[],[[[]]]]] => 5

[[],[[[]],[],[]]] => 3

[[],[[[]],[[]]]] => 4

[[],[[[],[]],[]]] => 3

[[],[[[[]]],[]]] => 3

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$F_{2} = q$

$F_{3} = 2\ q^{2}$

$F_{4} = q^{2} + 4\ q^{3}$

$F_{5} = 2\ q^{2} + 5\ q^{3} + 7\ q^{4}$

$F_{6} = 5\ q^{2} + 12\ q^{3} + 14\ q^{4} + 11\ q^{5}$

$F_{7} = 14\ q^{2} + 33\ q^{3} + 39\ q^{4} + 30\ q^{5} + 16\ q^{6}$

$F_{8} = 42\ q^{2} + 98\ q^{3} + 117\ q^{4} + 95\ q^{5} + 55\ q^{6} + 22\ q^{7}$

Description

The length of the boundary minus the length of the trunk of an ordered tree.
This is the size of the set of edges which are either on the left most path or on the right most path from the root.

Code

def statistic(T):
    b = 0
    C = T
    while len(C) > 0:
        C = C[0]
        b += 1
    C = T
    while len(C) > 0:
        C = C[-1]
        b += 1
    while len(T) == 1:
        T = T[0]
        b -= 1
    return b

Created

Sep 06, 2017 at 23:35 by Martin Rubey

Updated

Sep 06, 2017 at 23:35 by Martin Rubey