- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000078: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1

[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4

[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 3

[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,3,4] => 6

[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,3,4] => 6

[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 6

[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2

[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 13

[1,3,4,2,5] => [1,4,3,2,5] => [1,4,2,5,3] => 8

[1,3,4,5,2] => [1,5,3,4,2] => [1,5,2,3,4] => 4

[1,3,5,2,4] => [1,4,5,2,3] => [1,4,5,2,3] => 6

[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[1,4,2,3,5] => [1,4,3,2,5] => [1,4,2,5,3] => 8

[1,4,2,5,3] => [1,5,3,4,2] => [1,5,2,3,4] => 4

[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,5,3] => 8

[1,4,3,5,2] => [1,5,3,4,2] => [1,5,2,3,4] => 4

[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[1,4,5,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[1,5,2,3,4] => [1,5,3,4,2] => [1,5,2,3,4] => 4

[1,5,2,4,3] => [1,5,3,4,2] => [1,5,2,3,4] => 4

[1,5,3,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[1,5,3,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[1,5,4,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 4

[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => 4

[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,2,4] => 8

[2,1,4,3,5] => [2,1,4,3,5] => [1,4,2,3,5] => 3

[2,1,4,5,3] => [2,1,5,4,3] => [1,5,2,3,4] => 4

[2,1,5,3,4] => [2,1,5,4,3] => [1,5,2,3,4] => 4

[2,1,5,4,3] => [2,1,5,4,3] => [1,5,2,3,4] => 4

[2,3,1,4,5] => [3,2,1,4,5] => [1,4,5,2,3] => 6

[2,3,1,5,4] => [3,2,1,5,4] => [1,5,2,3,4] => 4

[2,3,4,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => 4

[2,3,4,5,1] => [5,2,3,4,1] => [1,2,3,4,5] => 1

[2,3,5,1,4] => [4,2,5,1,3] => [1,3,2,5,4] => 13

[2,3,5,4,1] => [5,2,4,3,1] => [1,2,4,3,5] => 3

[2,4,1,3,5] => [3,4,1,2,5] => [1,2,5,3,4] => 6

[2,4,1,5,3] => [3,5,1,4,2] => [1,4,2,3,5] => 3

[2,4,3,1,5] => [4,3,2,1,5] => [1,5,2,3,4] => 4

[2,4,3,5,1] => [5,3,2,4,1] => [1,2,4,3,5] => 3

[2,4,5,1,3] => [4,5,3,1,2] => [1,2,3,4,5] => 1

[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1

[2,5,1,3,4] => [3,5,1,4,2] => [1,4,2,3,5] => 3

[2,5,1,4,3] => [3,5,1,4,2] => [1,4,2,3,5] => 3

[2,5,3,1,4] => [4,5,3,1,2] => [1,2,3,4,5] => 1

[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1

[2,5,4,1,3] => [4,5,3,1,2] => [1,2,3,4,5] => 1

[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1

[3,1,2,4,5] => [3,2,1,4,5] => [1,4,5,2,3] => 6

[3,1,2,5,4] => [3,2,1,5,4] => [1,5,2,3,4] => 4

[3,1,4,2,5] => [4,2,3,1,5] => [1,5,2,3,4] => 4

[3,1,4,5,2] => [5,2,3,4,1] => [1,2,3,4,5] => 1

[3,1,5,2,4] => [4,2,5,1,3] => [1,3,2,5,4] => 13

[3,1,5,4,2] => [5,2,4,3,1] => [1,2,4,3,5] => 3

[3,2,1,4,5] => [3,2,1,4,5] => [1,4,5,2,3] => 6

[3,2,1,5,4] => [3,2,1,5,4] => [1,5,2,3,4] => 4

[3,2,4,1,5] => [4,2,3,1,5] => [1,5,2,3,4] => 4

[3,2,4,5,1] => [5,2,3,4,1] => [1,2,3,4,5] => 1

[3,2,5,1,4] => [4,2,5,1,3] => [1,3,2,5,4] => 13

[3,2,5,4,1] => [5,2,4,3,1] => [1,2,4,3,5] => 3

[3,4,1,2,5] => [4,3,2,1,5] => [1,5,2,3,4] => 4

[3,4,1,5,2] => [5,3,2,4,1] => [1,2,4,3,5] => 3

[3,4,2,1,5] => [4,3,2,1,5] => [1,5,2,3,4] => 4

[3,4,2,5,1] => [5,3,2,4,1] => [1,2,4,3,5] => 3

[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1

[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1

[3,5,1,2,4] => [4,5,3,1,2] => [1,2,3,4,5] => 1

[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1

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Description

The number of alternating sign matrices whose left key is the permutation.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.

Map

Demazure product with inverse

Description

This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.

Map

runsort

Description

The permutation obtained by sorting the increasing runs lexicographically.