- FindStat

Identifier

Mp00048: Ordered trees —left-right symmetry⟶ Ordered trees
St000400: Ordered trees ⟶ ℤ

Values

[[]] => [[]] => 1

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

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

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

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

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

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

[[[[]]]] => [[[[]]]] => 6

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

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

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

[[],[[],[]]] => [[[],[]],[]] => 6

[[],[[[]]]] => [[[[]]],[]] => 7

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

[[[]],[[]]] => [[[]],[[]]] => 6

[[[],[]],[]] => [[],[[],[]]] => 6

[[[[]]],[]] => [[],[[[]]]] => 7

[[[],[],[]]] => [[[],[],[]]] => 7

[[[],[[]]]] => [[[[]],[]]] => 8

[[[[]],[]]] => [[[],[[]]]] => 8

[[[[],[]]]] => [[[[],[]]]] => 9

[[[[[]]]]] => [[[[[]]]]] => 10

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

[[],[],[],[[]]] => [[[]],[],[],[]] => 6

[[],[],[[]],[]] => [[],[[]],[],[]] => 6

[[],[],[[],[]]] => [[[],[]],[],[]] => 7

[[],[],[[[]]]] => [[[[]]],[],[]] => 8

[[],[[]],[],[]] => [[],[],[[]],[]] => 6

[[],[[]],[[]]] => [[[]],[[]],[]] => 7

[[],[[],[]],[]] => [[],[[],[]],[]] => 7

[[],[[[]]],[]] => [[],[[[]]],[]] => 8

[[],[[],[],[]]] => [[[],[],[]],[]] => 8

[[],[[],[[]]]] => [[[[]],[]],[]] => 9

[[],[[[]],[]]] => [[[],[[]]],[]] => 9

[[],[[[],[]]]] => [[[[],[]]],[]] => 10

[[],[[[[]]]]] => [[[[[]]]],[]] => 11

[[[]],[],[],[]] => [[],[],[],[[]]] => 6

[[[]],[],[[]]] => [[[]],[],[[]]] => 7

[[[]],[[]],[]] => [[],[[]],[[]]] => 7

[[[]],[[],[]]] => [[[],[]],[[]]] => 8

[[[]],[[[]]]] => [[[[]]],[[]]] => 9

[[[],[]],[],[]] => [[],[],[[],[]]] => 7

[[[[]]],[],[]] => [[],[],[[[]]]] => 8

[[[],[]],[[]]] => [[[]],[[],[]]] => 8

[[[[]]],[[]]] => [[[]],[[[]]]] => 9

[[[],[],[]],[]] => [[],[[],[],[]]] => 8

[[[],[[]]],[]] => [[],[[[]],[]]] => 9

[[[[]],[]],[]] => [[],[[],[[]]]] => 9

[[[[],[]]],[]] => [[],[[[],[]]]] => 10

[[[[[]]]],[]] => [[],[[[[]]]]] => 11

[[[],[],[],[]]] => [[[],[],[],[]]] => 9

[[[],[],[[]]]] => [[[[]],[],[]]] => 10

[[[],[[]],[]]] => [[[],[[]],[]]] => 10

[[[],[[],[]]]] => [[[[],[]],[]]] => 11

[[[],[[[]]]]] => [[[[[]]],[]]] => 12

[[[[]],[],[]]] => [[[],[],[[]]]] => 10

[[[[]],[[]]]] => [[[[]],[[]]]] => 11

[[[[],[]],[]]] => [[[],[[],[]]]] => 11

[[[[[]]],[]]] => [[[],[[[]]]]] => 12

[[[[],[],[]]]] => [[[[],[],[]]]] => 12

[[[[],[[]]]]] => [[[[[]],[]]]] => 13

[[[[[]],[]]]] => [[[[],[[]]]]] => 13

[[[[[],[]]]]] => [[[[[],[]]]]] => 14

[[[[[[]]]]]] => [[[[[[]]]]]] => 15

[[],[],[],[],[],[]] => [[],[],[],[],[],[]] => 6

[[],[],[],[],[[]]] => [[[]],[],[],[],[]] => 7

[[],[],[],[[]],[]] => [[],[[]],[],[],[]] => 7

[[],[],[],[[],[]]] => [[[],[]],[],[],[]] => 8

[[],[],[],[[[]]]] => [[[[]]],[],[],[]] => 9

[[],[],[[]],[],[]] => [[],[],[[]],[],[]] => 7

[[],[],[[]],[[]]] => [[[]],[[]],[],[]] => 8

[[],[],[[],[]],[]] => [[],[[],[]],[],[]] => 8

[[],[],[[[]]],[]] => [[],[[[]]],[],[]] => 9

[[],[],[[],[],[]]] => [[[],[],[]],[],[]] => 9

[[],[],[[],[[]]]] => [[[[]],[]],[],[]] => 10

[[],[],[[[]],[]]] => [[[],[[]]],[],[]] => 10

[[],[],[[[],[]]]] => [[[[],[]]],[],[]] => 11

[[],[],[[[[]]]]] => [[[[[]]]],[],[]] => 12

[[],[[]],[],[],[]] => [[],[],[],[[]],[]] => 7

[[],[[]],[],[[]]] => [[[]],[],[[]],[]] => 8

[[],[[]],[[]],[]] => [[],[[]],[[]],[]] => 8

[[],[[]],[[],[]]] => [[[],[]],[[]],[]] => 9

[[],[[]],[[[]]]] => [[[[]]],[[]],[]] => 10

[[],[[],[]],[],[]] => [[],[],[[],[]],[]] => 8

[[],[[[]]],[],[]] => [[],[],[[[]]],[]] => 9

[[],[[],[]],[[]]] => [[[]],[[],[]],[]] => 9

[[],[[[]]],[[]]] => [[[]],[[[]]],[]] => 10

[[],[[],[],[]],[]] => [[],[[],[],[]],[]] => 9

[[],[[],[[]]],[]] => [[],[[[]],[]],[]] => 10

[[],[[[]],[]],[]] => [[],[[],[[]]],[]] => 10

[[],[[[],[]]],[]] => [[],[[[],[]]],[]] => 11

[[],[[[[]]]],[]] => [[],[[[[]]]],[]] => 12

[[],[[],[],[],[]]] => [[[],[],[],[]],[]] => 10

[[],[[],[],[[]]]] => [[[[]],[],[]],[]] => 11

[[],[[],[[]],[]]] => [[[],[[]],[]],[]] => 11

[[],[[],[[],[]]]] => [[[[],[]],[]],[]] => 12

[[],[[],[[[]]]]] => [[[[[]]],[]],[]] => 13

[[],[[[]],[],[]]] => [[[],[],[[]]],[]] => 11

[[],[[[]],[[]]]] => [[[[]],[[]]],[]] => 12

[[],[[[],[]],[]]] => [[[],[[],[]]],[]] => 12

[[],[[[[]]],[]]] => [[[],[[[]]]],[]] => 13

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$F_{2} = q$

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

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

$F_{5} = q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = q^{5} + 4\ q^{6} + 6\ q^{7} + 7\ q^{8} + 7\ q^{9} + 5\ q^{10} + 5\ q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

$F_{7} = q^{6} + 5\ q^{7} + 10\ q^{8} + 14\ q^{9} + 17\ q^{10} + 16\ q^{11} + 16\ q^{12} + 14\ q^{13} + 11\ q^{14} + 9\ q^{15} + 7\ q^{16} + 5\ q^{17} + 3\ q^{18} + 2\ q^{19} + q^{20} + q^{21}$

$F_{8} = q^{7} + 6\ q^{8} + 15\ q^{9} + 25\ q^{10} + 35\ q^{11} + 40\ q^{12} + 43\ q^{13} + 44\ q^{14} + 40\ q^{15} + 37\ q^{16} + 32\ q^{17} + 28\ q^{18} + 22\ q^{19} + 18\ q^{20} + 13\ q^{21} + 11\ q^{22} + 7\ q^{23} + 5\ q^{24} + 3\ q^{25} + 2\ q^{26} + q^{27} + q^{28}$

Description

The path length of an ordered tree.
This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].

Map

left-right symmetry

Description

Return the left-right symmetrized tree of an ordered tree.