- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,0] => [4] => 5

[1,1,0,1,1,0,0,0] => [4] => 5

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

[1,1,1,0,0,1,0,0] => [4] => 5

[1,1,1,0,1,0,0,0] => [4] => 5

[1,1,1,1,0,0,0,0] => [4] => 5

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

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

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

[1,0,1,0,1,1,0,1,0,0] => [1,1,3] => 6

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 6

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

[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] => 6

[1,0,1,1,0,1,0,1,0,0] => [1,4] => 6

[1,0,1,1,0,1,1,0,0,0] => [1,4] => 6

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 6

[1,0,1,1,1,0,0,1,0,0] => [1,4] => 6

[1,0,1,1,1,0,1,0,0,0] => [1,4] => 6

[1,0,1,1,1,1,0,0,0,0] => [1,4] => 6

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

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

[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] => 5

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

[1,1,0,1,0,0,1,0,1,0] => [3,1,1] => 6

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

[1,1,0,1,0,1,0,0,1,0] => [4,1] => 6

[1,1,0,1,0,1,0,1,0,0] => [5] => 6

[1,1,0,1,0,1,1,0,0,0] => [5] => 6

[1,1,0,1,1,0,0,0,1,0] => [4,1] => 6

[1,1,0,1,1,0,0,1,0,0] => [5] => 6

[1,1,0,1,1,0,1,0,0,0] => [5] => 6

[1,1,0,1,1,1,0,0,0,0] => [5] => 6

[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 6

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

[1,1,1,0,0,1,0,0,1,0] => [4,1] => 6

[1,1,1,0,0,1,0,1,0,0] => [5] => 6

[1,1,1,0,0,1,1,0,0,0] => [5] => 6

[1,1,1,0,1,0,0,0,1,0] => [4,1] => 6

[1,1,1,0,1,0,0,1,0,0] => [5] => 6

[1,1,1,0,1,0,1,0,0,0] => [5] => 6

[1,1,1,0,1,1,0,0,0,0] => [5] => 6

[1,1,1,1,0,0,0,0,1,0] => [4,1] => 6

[1,1,1,1,0,0,0,1,0,0] => [5] => 6

[1,1,1,1,0,0,1,0,0,0] => [5] => 6

[1,1,1,1,0,1,0,0,0,0] => [5] => 6

[1,1,1,1,1,0,0,0,0,0] => [5] => 6

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => 7

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => 7

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => 7

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => 7

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => 7

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => 7

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => 7

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => 7

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => 7

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => 7

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => 7

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => 7

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 7

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => 7

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => 7

[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] => 6

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => 6

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => 7

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => 6

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => 7

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => 7

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => 7

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => 7

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => 7

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => 7

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => 7

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => 7

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => 6

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => 7

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => 7

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => 7

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => 7

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => 7

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => 7

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => 7

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => 7

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

$F_{3} = 5\ q^{4}$

$F_{4} = q^{4} + 13\ q^{5}$

$F_{5} = 6\ q^{5} + 36\ q^{6}$

$F_{6} = 5\ q^{5} + 21\ q^{6} + 106\ q^{7}$

$F_{7} = 36\ q^{6} + 66\ q^{7} + 327\ q^{8}$

$F_{8} = 34\ q^{6} + 142\ q^{7} + 209\ q^{8} + 1045\ q^{9}$

Description

The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.

Map

touch composition

Description

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