- 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

>>> Load all 2461 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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

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).

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