Identifier
Values
[1,0] => [[1],[2]] => [2,1] => [2,1] => 0
[1,0,1,0] => [[1,3],[2,4]] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,0,0] => [[1,2],[3,4]] => [3,4,1,2] => [3,1,4,2] => 0
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [6,4,2,1,3,5] => 4
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [5,2,1,6,3,4] => 1
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [6,3,1,4,2,5] => 3
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [5,1,6,3,2,4] => 1
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,4,6,8,1,3,5,7] => [8,6,4,2,1,3,5,7] => 9
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 0
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Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.
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
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.