Identifier
Values
0 => [2] => [1,1,0,0] => [[1,2],[3,4]] => 0
1 => [1,1] => [1,0,1,0] => [[1,3],[2,4]] => 1
00 => [3] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 0
01 => [2,1] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
10 => [1,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
000 => [4] => [1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 3
=> [1] => [1,0] => [[1],[2]] => 0
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Description
The number of natural descents 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.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
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.