- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 15

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [2,1,1,1,1] => 14

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => [1,2,1,1,1] => 13

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,2,1] => [1,2,1,1,1] => 13

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [3,1,1,1] => 12

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [1,1,2,1,1] => 12

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

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,2,1,1] => [1,1,2,1,1] => 12

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,2,1,1] => [1,1,2,1,1] => 12

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see Permutations/Descents-Major.

Map

reverse

Description

Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.