- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{(2, 2)} = 2 + q$

$F_{(2, 3)} = 3 + 2\ q$

$F_{(3, 2)} = 3 + 2\ q$

$F_{(2, 4)} = 4 + 3\ q$

$F_{(3, 3)} = 6 + 6\ q + q^{3}$

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

$F_{(2, 5)} = 5 + 4\ q$

$F_{(3, 4)} = 10 + 12\ q + 3\ q^{3}$

$F_{(4, 3)} = 10 + 12\ q + 5\ q^{2} + 3\ q^{3}$

$F_{(5, 2)} = 5 + 4\ q + 2\ q^{2}$

$F_{(2, 6)} = 6 + 5\ q$

$F_{(3, 5)} = 15 + 20\ q + 6\ q^{3}$

$F_{(4, 4)} = 20 + 30\ q + 14\ q^{2} + 12\ q^{3} + q^{6}$

$F_{(5, 3)} = 15 + 20\ q + 13\ q^{2} + 6\ q^{3} + 3\ q^{4}$

$F_{(6, 2)} = 6 + 5\ q + 3\ q^{2} + q^{3}$

$F_{(2, 7)} = 7 + 6\ q$

$F_{(3, 6)} = 21 + 30\ q + 10\ q^{3}$

$F_{(4, 5)} = 35 + 60\ q + 30\ q^{2} + 30\ q^{3} + 4\ q^{6}$

$F_{(5, 4)} = 35 + 60\ q + 45\ q^{2} + 30\ q^{3} + 17\ q^{4} + 4\ q^{6}$

$F_{(6, 3)} = 21 + 30\ q + 24\ q^{2} + 19\ q^{3} + 8\ q^{4} + q^{6}$

$F_{(7, 2)} = 7 + 6\ q + 4\ q^{2} + 2\ q^{3}$

$F_{(2, 8)} = 8 + 7\ q$

$F_{(3, 7)} = 28 + 42\ q + 15\ q^{3}$

$F_{(4, 6)} = 56 + 105\ q + 55\ q^{2} + 60\ q^{3} + 10\ q^{6}$

$F_{(5, 5)} = 70 + 140\ q + 115\ q^{2} + 90\ q^{3} + 55\ q^{4} + 20\ q^{6} + q^{10}$

$F_{(6, 4)} = 56 + 105\ q + 99\ q^{2} + 100\ q^{3} + 56\ q^{4} + 19\ q^{6} + 6\ q^{7}$

$F_{(7, 3)} = 28 + 42\ q + 38\ q^{2} + 37\ q^{3} + 15\ q^{4} + 6\ q^{5} + 3\ q^{6}$

$F_{(8, 2)} = 8 + 7\ q + 5\ q^{2} + 3\ q^{3} + q^{4}$

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

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation

$\pi$ of length

$n$ is the integer composition of

$n$ whose descent set equals the descent set of

$\pi$ . The descent set of a permutation

$\pi$ is

$\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$ . The descent set of a composition

$c = (i_1, i_2, \ldots, i_k)$ is the set

$\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$ .

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).