- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions

Images

[[1,2]] => [1,2] => [1,2] => {{1},{2}}

[[2,2]] => [1,2] => [1,2] => {{1},{2}}

[[1],[2]] => [2,1] => [2,1] => {{1,2}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2458 entries. <<<

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.

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.