Identifier
Values
[1,0] => [1,0] => [[1],[2]] => 0
[1,0,1,0] => [1,1,0,0] => [[1,2],[3,4]] => 0
[1,1,0,0] => [1,0,1,0] => [[1,3],[2,4]] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 4
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 2
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 6
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 0
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 6
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 4
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 5
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 10
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 8
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 4
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 9
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 6
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 7
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 8
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 12
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Description
The major index of a standard tableau minus the weighted size of its shape.
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.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.