- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001941: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,3,4,2,5] => [1,4,3,2,5] => [5,4,3,2,1] => 23

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

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

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

[1,4,2,3,5] => [1,4,3,2,5] => [5,4,3,2,1] => 23

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

[1,4,3,2,5] => [1,4,3,2,5] => [5,4,3,2,1] => 23

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,4,3,1,5] => [4,3,2,1,5] => [5,4,3,1,2] => 14

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,1,2] => 14

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

[3,4,2,1,5] => [4,3,2,1,5] => [5,4,3,1,2] => 14

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

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

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

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

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

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Description

The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5.3]) with parameters given by the identity and the permutation.
Also the number of paths in the Bruhat order from the identity to the permutation that are increasing with respect to a given reflection ordering as defined in Björner and Brenti [1, Section 5.2].

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

inverse toric promotion

Description

Toric promotion of a permutation.
Let $\sigma\in\mathfrak S_n$ be a permutation and let
$ \tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases} $
The toric promotion operator is the product $\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$.
This is the special case of toric promotion on graphs for the path graph. Its order is $n-1$.