- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤ

Values

[[]] => [1,0] => [1] => 1

[[],[]] => [1,0,1,0] => [1,1] => 2

[[[]]] => [1,1,0,0] => [2] => 2

[[],[],[]] => [1,0,1,0,1,0] => [1,1,1] => 2

[[],[[]]] => [1,0,1,1,0,0] => [1,2] => 3

[[[]],[]] => [1,1,0,0,1,0] => [2,1] => 3

[[[],[]]] => [1,1,0,1,0,0] => [3] => 2

[[[[]]]] => [1,1,1,0,0,0] => [3] => 2

[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => 2

[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,1,2] => 3

[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,2,1] => 4

[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,3] => 3

[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,3] => 3

[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,1] => 3

[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,2] => 4

[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [3,1] => 3

[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [3,1] => 3

[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [4] => 2

[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [4] => 2

[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [4] => 2

[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [4] => 2

[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4] => 2

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => 2

[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => 3

[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => 4

[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => 3

[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 3

[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => 4

[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,2,2] => 5

[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,3,1] => 4

[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 4

[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,4] => 3

[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,4] => 3

[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,4] => 3

[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,4] => 3

[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 3

[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => 3

[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,2] => 4

[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,2,1] => 5

[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,3] => 4

[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => 4

[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [3,1,1] => 3

[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 3

[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [3,2] => 4

[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => 4

[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [4,1] => 3

[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [4,1] => 3

[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [4,1] => 3

[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 3

[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,1] => 3

[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [5] => 2

[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [5] => 2

[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [5] => 2

[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [5] => 2

[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [5] => 2

[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [5] => 2

[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [5] => 2

[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [5] => 2

[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [5] => 2

[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [5] => 2

[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5] => 2

[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [5] => 2

[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [5] => 2

[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5] => 2

[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => 2

[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => 3

[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => 4

[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => 3

[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => 3

[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => 4

[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => 5

[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => 4

[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => 4

[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => 3

[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => 3

[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => 3

[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => 3

[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 3

[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => 4

[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => 5

[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => 6

[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,3] => 5

[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => 5

[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => 4

[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => 4

[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => 5

[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => 5

[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => 4

[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => 4

[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => 4

[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => 4

[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => 4

[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => 3

[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => 3

[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => 3

[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => 3

[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => 3

[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => 3

[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => 3

[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => 3

[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5] => 3

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Description

The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.