- FindStat

Identifier

Mp00295: Standard tableaux —valley composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5]] => [5] => 0

[[1,3,4,5],[2]] => [2,3] => 2

[[1,2,4,5],[3]] => [3,2] => 3

[[1,2,3,5],[4]] => [4,1] => 4

[[1,2,3,4],[5]] => [5] => 0

[[1,3,5],[2,4]] => [2,2,1] => 6

[[1,2,5],[3,4]] => [3,2] => 3

[[1,3,4],[2,5]] => [2,3] => 2

[[1,2,4],[3,5]] => [3,2] => 3

[[1,2,3],[4,5]] => [4,1] => 4

[[1,4,5],[2],[3]] => [3,2] => 3

[[1,3,5],[2],[4]] => [2,2,1] => 6

[[1,2,5],[3],[4]] => [4,1] => 4

[[1,3,4],[2],[5]] => [2,3] => 2

[[1,2,4],[3],[5]] => [3,2] => 3

[[1,2,3],[4],[5]] => [5] => 0

[[1,4],[2,5],[3]] => [3,2] => 3

[[1,3],[2,5],[4]] => [2,2,1] => 6

[[1,2],[3,5],[4]] => [4,1] => 4

[[1,3],[2,4],[5]] => [2,3] => 2

[[1,2],[3,4],[5]] => [3,2] => 3

[[1,5],[2],[3],[4]] => [4,1] => 4

[[1,4],[2],[3],[5]] => [3,2] => 3

[[1,3],[2],[4],[5]] => [2,3] => 2

[[1,2],[3],[4],[5]] => [5] => 0

[[1],[2],[3],[4],[5]] => [5] => 0

[[1,2,3,4,5,6]] => [6] => 0

[[1,3,4,5,6],[2]] => [2,4] => 2

[[1,2,4,5,6],[3]] => [3,3] => 3

[[1,2,3,5,6],[4]] => [4,2] => 4

[[1,2,3,4,6],[5]] => [5,1] => 5

[[1,2,3,4,5],[6]] => [6] => 0

[[1,3,5,6],[2,4]] => [2,2,2] => 6

[[1,2,5,6],[3,4]] => [3,3] => 3

[[1,3,4,6],[2,5]] => [2,3,1] => 7

[[1,2,4,6],[3,5]] => [3,2,1] => 8

[[1,2,3,6],[4,5]] => [4,2] => 4

[[1,3,4,5],[2,6]] => [2,4] => 2

[[1,2,4,5],[3,6]] => [3,3] => 3

[[1,2,3,5],[4,6]] => [4,2] => 4

[[1,2,3,4],[5,6]] => [5,1] => 5

[[1,4,5,6],[2],[3]] => [3,3] => 3

[[1,3,5,6],[2],[4]] => [2,2,2] => 6

[[1,2,5,6],[3],[4]] => [4,2] => 4

[[1,3,4,6],[2],[5]] => [2,3,1] => 7

[[1,2,4,6],[3],[5]] => [3,2,1] => 8

[[1,2,3,6],[4],[5]] => [5,1] => 5

[[1,3,4,5],[2],[6]] => [2,4] => 2

[[1,2,4,5],[3],[6]] => [3,3] => 3

[[1,2,3,5],[4],[6]] => [4,2] => 4

[[1,2,3,4],[5],[6]] => [6] => 0

[[1,3,5],[2,4,6]] => [2,2,2] => 6

[[1,2,5],[3,4,6]] => [3,3] => 3

[[1,3,4],[2,5,6]] => [2,3,1] => 7

[[1,2,4],[3,5,6]] => [3,2,1] => 8

[[1,2,3],[4,5,6]] => [4,2] => 4

[[1,4,6],[2,5],[3]] => [3,2,1] => 8

[[1,3,6],[2,5],[4]] => [2,2,2] => 6

[[1,2,6],[3,5],[4]] => [4,2] => 4

[[1,3,6],[2,4],[5]] => [2,3,1] => 7

[[1,2,6],[3,4],[5]] => [3,2,1] => 8

[[1,4,5],[2,6],[3]] => [3,3] => 3

[[1,3,5],[2,6],[4]] => [2,2,2] => 6

[[1,2,5],[3,6],[4]] => [4,2] => 4

[[1,3,4],[2,6],[5]] => [2,3,1] => 7

[[1,2,4],[3,6],[5]] => [3,2,1] => 8

[[1,2,3],[4,6],[5]] => [5,1] => 5

[[1,3,5],[2,4],[6]] => [2,2,2] => 6

[[1,2,5],[3,4],[6]] => [3,3] => 3

[[1,3,4],[2,5],[6]] => [2,4] => 2

[[1,2,4],[3,5],[6]] => [3,3] => 3

[[1,2,3],[4,5],[6]] => [4,2] => 4

[[1,5,6],[2],[3],[4]] => [4,2] => 4

[[1,4,6],[2],[3],[5]] => [3,2,1] => 8

[[1,3,6],[2],[4],[5]] => [2,3,1] => 7

[[1,2,6],[3],[4],[5]] => [5,1] => 5

[[1,4,5],[2],[3],[6]] => [3,3] => 3

[[1,3,5],[2],[4],[6]] => [2,2,2] => 6

[[1,2,5],[3],[4],[6]] => [4,2] => 4

[[1,3,4],[2],[5],[6]] => [2,4] => 2

[[1,2,4],[3],[5],[6]] => [3,3] => 3

[[1,2,3],[4],[5],[6]] => [6] => 0

[[1,4],[2,5],[3,6]] => [3,3] => 3

[[1,3],[2,5],[4,6]] => [2,2,2] => 6

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$F_{1} = 1$

$F_{2} = 2$

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

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

$F_{5} = 5 + 5\ q^{2} + 8\ q^{3} + 5\ q^{4} + 3\ q^{6}$

$F_{6} = 6 + 7\ q^{2} + 15\ q^{3} + 15\ q^{4} + 7\ q^{5} + 9\ q^{6} + 8\ q^{7} + 9\ q^{8}$

$F_{7} = 7 + 9\ q^{2} + 22\ q^{3} + 29\ q^{4} + 22\ q^{5} + 26\ q^{6} + 24\ q^{7} + 41\ q^{8} + 24\ q^{9} + 17\ q^{10} + 11\ q^{12}$

$F_{8} = 8 + 11\ q^{2} + 29\ q^{3} + 46\ q^{4} + 46\ q^{5} + 55\ q^{6} + 60\ q^{7} + 100\ q^{8} + 98\ q^{9} + 100\ q^{10} + 49\ q^{11} + 63\ q^{12} + 31\ q^{13} + 31\ q^{14} + 37\ q^{15}$

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

valley composition

Description

The composition corresponding to the valley set of a standard tableau.
Let

$T$ be a standard tableau of size

$n$ .
An entry

$i$ of

$T$ is a descent if

$i+1$ is in a lower row (in English notation), otherwise

$i$ is an ascent.
An entry

$2 \leq i \leq n-1$ is a valley if

$i-1$ is a descent and

$i$ is an ascent.
This map returns the composition

$c_1,\dots,c_k$ of

$n$ such that

$\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of

$T$ .