- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000062: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The length of the longest increasing subsequence of the permutation.

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

Clarke-Steingrimsson-Zeng

Description

The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where

$des$ is the number of descents, St000021The number of descents of a permutation.,
$exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
$Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
$Ebot$ is the sum of the excedance bottoms,
$Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
$Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
$Res$ is the sum of the (right) embracing numbers,
$Ine$ is the sum of the side numbers.