- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000436: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2278 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 number of occurrences of the pattern 231 or of the pattern 321 in a 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

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.