- FindStat

Identifier

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000371: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also St000119The number of occurrences 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).