- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000769: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 10 + 3\ q + q^{2}$

$F_{5} = 26 + 11\ q + 4\ q^{2} + q^{3}$

$F_{6} = 78 + 32\ q + 14\ q^{2} + 6\ q^{3} + 2\ q^{4}$

$F_{7} = 236 + 110\ q + 47\ q^{2} + 22\ q^{3} + 9\ q^{4} + 4\ q^{5} + q^{6}$

$F_{8} = 770 + 347\ q + 160\ q^{2} + 83\ q^{3} + 39\ q^{4} + 19\ q^{5} + 8\ q^{6} + 3\ q^{7} + q^{8}$

Description

The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see St000008The major index of the composition..

Map

touch composition

Description

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