- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001213: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,1,0,0] => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0] => 7

[1,0,1,1,1,0,0,0] => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0] => 7

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

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

[1,1,0,1,0,0,1,0] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 7

[1,1,0,1,0,1,0,0] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 8

[1,1,0,1,1,0,0,0] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 8

[1,1,1,0,0,0,1,0] => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 7

[1,1,1,0,0,1,0,0] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 8

[1,1,1,0,1,0,0,0] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 8

[1,1,1,1,0,0,0,0] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 8

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

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 7

[1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 7

[1,0,1,0,1,1,0,1,0,0] => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 8

[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 8

[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 7

[1,0,1,1,0,0,1,1,0,0] => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 8

[1,0,1,1,0,1,0,0,1,0] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 8

[1,0,1,1,0,1,0,1,0,0] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,1,0,1,1,0,0,0] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 8

[1,0,1,1,1,0,0,1,0,0] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,1,1,0,1,0,0,0] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,0,1,1,1,1,0,0,0,0] => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 9

[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 7

[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 8

[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 8

[1,1,0,0,1,1,0,1,0,0] => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 9

[1,1,0,0,1,1,1,0,0,0] => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 9

[1,1,0,1,0,0,1,0,1,0] => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 8

[1,1,0,1,0,0,1,1,0,0] => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 9

[1,1,0,1,0,1,0,0,1,0] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,0,1,0,1,0,1,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,0,1,1,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,1,0,0,0,1,0] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,0,1,1,0,0,1,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,1,0,1,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,0,1,1,1,0,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 8

[1,1,1,0,0,0,1,1,0,0] => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 9

[1,1,1,0,0,1,0,0,1,0] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,1,0,0,1,0,1,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,0,1,1,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,1,0,1,0,0,1,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,1,0,1,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,0,1,1,0,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,1,0,0,0,0,1,0] => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9

[1,1,1,1,0,0,0,1,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,1,0,0,1,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,1,0,1,0,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,1,1,1,1,0,0,0,0,0] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 7

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 8

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 8

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 9

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 9

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 8

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 9

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 9

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 9

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 10

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 8

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 9

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 9

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 10

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 9

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 9

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 10

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 11

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$F_{1} = q^{2}$

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

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

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

$F_{5} = q^{6} + 4\ q^{7} + 9\ q^{8} + 14\ q^{9} + 14\ q^{10}$

$F_{6} = q^{7} + 5\ q^{8} + 14\ q^{9} + 28\ q^{10} + 42\ q^{11} + 42\ q^{12}$

Description

The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

conjugate

Description

The conjugate of a composition.
The conjugate of a composition

$C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of

$C$ .
Equivalently, the ribbon shape corresponding to the conjugate of

$C$ is the conjugate of the ribbon shape of

$C$ .

Map

touch composition

Description

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