Identifier
Values
[1] => [[1]] => [1] => [1] => 0
[2] => [[1,2]] => [1,2] => [1,2] => 0
[1,1] => [[1],[2]] => [2,1] => [1,2] => 0
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => 0
[2,1] => [[1,3],[2]] => [2,1,3] => [1,3,2] => 0
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [1,2,3] => 0
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [1,3,4,2] => 0
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [1,2,3,4] => 0
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [1,4,2,3] => 0
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [1,2,3,4] => 0
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [1,3,4,5,2] => 0
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [1,2,5,3,4] => 0
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [1,4,5,2,3] => 0
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [1,3,2,5,4] => 0
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [1,5,2,3,4] => 0
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [1,2,5,6,3,4] => 0
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [1,4,5,6,2,3] => 0
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [1,3,6,2,5,4] => 1
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [1,5,6,2,3,4] => 0
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 0
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [1,4,2,6,3,5] => 0
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 0
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => 0
[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => 0
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [1,4,5,6,7,2,3] => 0
[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => 0
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => 1
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [1,5,6,7,2,3,4] => 0
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => [1,3,4,2,6,7,5] => 0
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => [1,2,7,3,4,5,6] => 0
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [1,6,7,2,3,4,5] => 0
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => 0
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => [1,5,2,7,3,4,6] => 0
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [1,7,2,3,4,5,6] => 0
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 0
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [1,2,3,4,5,6,7,8] => 0
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => [1,6,7,8,2,3,4,5] => 0
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => [1,7,8,2,3,4,5,6] => 0
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8] => 0
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4] => [1,4,2,6,3,8,5,7] => 0
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => [1,6,2,8,3,4,5,7] => 0
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => [1,8,2,3,4,5,6,7] => 0
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => [1,2,3,4,5,6,7,8,9] => 0
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => [1,8,9,2,3,4,5,6,7] => 0
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => [1,9,2,3,4,5,6,7,8] => 0
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => 0
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [6,7,8,9,10,1,2,3,4,5] => [1,2,3,4,5,6,7,8,9,10] => 0
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9,10] => [1,9,10,2,3,4,5,6,7,8] => 0
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [9,10,7,8,5,6,3,4,1,2] => [1,2,3,4,5,6,7,8,9,10] => 0
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1,10] => [1,10,2,3,4,5,6,7,8,9] => 0
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0
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
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 bottom-most row in English notation.
Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.