- FindStat

Identifier

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

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

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

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

[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

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

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

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

[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

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

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

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

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

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

[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

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

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

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

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

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

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

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

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

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

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

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

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

[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

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

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

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

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

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

[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

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

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

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

[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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 15\ q^{2} + 16\ q^{3} + 9\ q^{4} + 2\ q^{5}$

$F_{6} = 43\ q^{2} + 44\ q^{3} + 31\ q^{4} + 12\ q^{5} + 2\ q^{6}$

$F_{7} = 133\ q^{2} + 128\ q^{3} + 101\ q^{4} + 50\ q^{5} + 15\ q^{6} + 2\ q^{7}$

$F_{8} = 430\ q^{2} + 392\ q^{3} + 330\ q^{4} + 184\ q^{5} + 74\ q^{6} + 18\ q^{7} + 2\ q^{8}$

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

touch composition

Description

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