Identifier
Values
[1,0] => [1,0] => [1,0] => [[1],[2]] => 0
[1,0,1,0] => [1,0,1,0] => [1,1,0,0] => [[1,2],[3,4]] => 0
[1,1,0,0] => [1,1,0,0] => [1,0,1,0] => [[1,3],[2,4]] => 2
[1,0,1,0,1,0] => [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] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 4
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 2
[1,1,0,1,0,0] => [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] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 6
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [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] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 6
[1,0,1,1,0,0,1,0] => [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] => [1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 5
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 10
[1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 2
[1,1,0,0,1,1,0,0] => [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] => [1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 3
[1,1,0,1,0,1,0,0] => [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] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 9
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 6
[1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 7
[1,1,1,0,1,0,0,0] => [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] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 12
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Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
The natural comajor index of a tableau of size $n$ with natural descent set $D$ is then $\sum_{d\in D} n-d$.
Map
reverse
Description
The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.
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.
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.