Identifier
Values
[1,0] => [[1],[2]] => 1
[1,0,1,0] => [[1,3],[2,4]] => 4
[1,1,0,0] => [[1,2],[3,4]] => 2
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 9
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 7
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 5
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 6
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 16
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 14
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 12
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 13
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 10
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 10
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 8
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 11
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 12
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 9
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 6
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 7
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 8
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 4
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Description
The shifted 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 shape $\lambda$, size $n$ with natural descent set $D$ is then $b(\lambda)+\sum_{d\in D} n-d$, where $b(\lambda) = \sum_i (i-1)\lambda_i$.
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.