- FindStat

Identifier

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

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 10

[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] => 7

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

[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] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 15

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 13

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + 2\ q^{2} + q^{3}$

$F_{4} = 1 + q + 2\ q^{2} + 4\ q^{3} + 2\ q^{4} + 3\ q^{5} + q^{6}$

$F_{5} = 1 + q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 6\ q^{5} + 6\ q^{6} + 8\ q^{7} + 3\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 11\ q^{5} + 10\ q^{6} + 15\ q^{7} + 15\ q^{8} + 19\ q^{9} + 14\ q^{10} + 11\ q^{11} + 13\ q^{12} + 4\ q^{13} + 5\ q^{14} + q^{15}$

$F_{7} = 1 + q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 11\ q^{5} + 16\ q^{6} + 20\ q^{7} + 24\ q^{8} + 35\ q^{9} + 35\ q^{10} + 45\ q^{11} + 40\ q^{12} + 41\ q^{13} + 37\ q^{14} + 39\ q^{15} + 24\ q^{16} + 17\ q^{17} + 19\ q^{18} + 5\ q^{19} + 6\ q^{20} + q^{21}$

$F_{8} = 1 + q + 2\ q^{2} + 4\ q^{3} + 6\ q^{4} + 11\ q^{5} + 16\ q^{6} + 27\ q^{7} + 30\ q^{8} + 46\ q^{9} + 55\ q^{10} + 72\ q^{11} + 85\ q^{12} + 99\ q^{13} + 104\ q^{14} + 111\ q^{15} + 122\ q^{16} + 105\ q^{17} + 123\ q^{18} + 91\ q^{19} + 85\ q^{20} + 68\ q^{21} + 66\ q^{22} + 36\ q^{23} + 24\ q^{24} + 26\ q^{25} + 6\ q^{26} + 7\ q^{27} + q^{28}$

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

rise composition

Description

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