- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001004: Permutations ⟶ ℤ

Values

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

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

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

[[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] => 2

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

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

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

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

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

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

[[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see St000371The number of mid points of decreasing subsequences of length 3 in a permutation. and St000372The number of mid points of increasing subsequences of length 3 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

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 $n-1$.