Identifier
Mp00033:
Dyck paths
—to two-row standard tableau⟶
Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Images
[1,0] => [[1],[2]] => 1 => 1
[1,0,1,0] => [[1,3],[2,4]] => 101 => 101
[1,1,0,0] => [[1,2],[3,4]] => 010 => 010
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 10101 => 10101
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 10010 => 01001
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 01001 => 10010
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 01010 => 01010
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 00100 => 00100
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 1010101 => 1010101
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 1010010 => 0100101
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1001001 => 1001001
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 1001010 => 0101001
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 1000100 => 0010001
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 0100101 => 1010010
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 0100010 => 0100010
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 0101001 => 1001010
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 0101010 => 0101010
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 0100100 => 0010010
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 0010001 => 1000100
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 0010010 => 0100100
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 0010100 => 0010100
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 0001000 => 0001000
[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 101010101 => 101010101
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => 101010010 => 010010101
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => 101001001 => 100100101
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => 101001010 => 010100101
[1,0,1,0,1,1,1,0,0,0] => [[1,3,5,6,7],[2,4,8,9,10]] => 101000100 => 001000101
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 100100101 => 101001001
[1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => 100100010 => 010001001
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => 100101001 => 100101001
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => 100101010 => 010101001
[1,0,1,1,0,1,1,0,0,0] => [[1,3,4,6,7],[2,5,8,9,10]] => 100100100 => 001001001
[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => 100010001 => 100010001
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => 100010010 => 010010001
[1,0,1,1,1,0,1,0,0,0] => [[1,3,4,5,7],[2,6,8,9,10]] => 100010100 => 001010001
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => 100001000 => 000100001
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => 010010101 => 101010010
[1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => 010010010 => 010010010
[1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => 010001001 => 100100010
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => 010001010 => 010100010
[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => 010000100 => 001000010
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => 010100101 => 101001010
[1,1,0,1,0,0,1,1,0,0] => [[1,2,4,7,8],[3,5,6,9,10]] => 010100010 => 010001010
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => 010101001 => 100101010
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => 010101010 => 010101010
[1,1,0,1,0,1,1,0,0,0] => [[1,2,4,6,7],[3,5,8,9,10]] => 010100100 => 001001010
[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => 010010001 => 100010010
[1,1,0,1,1,0,0,1,0,0] => [[1,2,4,5,8],[3,6,7,9,10]] => 010010010 => 010010010
[1,1,0,1,1,0,1,0,0,0] => [[1,2,4,5,7],[3,6,8,9,10]] => 010010100 => 001010010
[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => 010001000 => 000100010
[1,1,1,0,0,0,1,0,1,0] => [[1,2,3,7,9],[4,5,6,8,10]] => 001000101 => 101000100
[1,1,1,0,0,0,1,1,0,0] => [[1,2,3,7,8],[4,5,6,9,10]] => 001000010 => 010000100
[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => 001001001 => 100100100
[1,1,1,0,0,1,0,1,0,0] => [[1,2,3,6,8],[4,5,7,9,10]] => 001001010 => 010100100
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => 001000100 => 001000100
[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => 001010001 => 100010100
[1,1,1,0,1,0,0,1,0,0] => [[1,2,3,5,8],[4,6,7,9,10]] => 001010010 => 010010100
[1,1,1,0,1,0,1,0,0,0] => [[1,2,3,5,7],[4,6,8,9,10]] => 001010100 => 001010100
[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => 001001000 => 000100100
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => 000100001 => 100001000
[1,1,1,1,0,0,0,1,0,0] => [[1,2,3,4,8],[5,6,7,9,10]] => 000100010 => 010001000
[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => 000100100 => 001001000
[1,1,1,1,0,1,0,0,0,0] => [[1,2,3,4,6],[5,7,8,9,10]] => 000101000 => 000101000
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => 000010000
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => 10101010101 => 10101010101
[1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => 10101010010 => 01001010101
[1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => 10101001001 => 10010010101
[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => 10101001010 => 01010010101
[1,0,1,0,1,0,1,1,1,0,0,0] => [[1,3,5,7,8,9],[2,4,6,10,11,12]] => 10101000100 => 00100010101
[1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => 10100100101 => 10100100101
[1,0,1,0,1,1,0,0,1,1,0,0] => [[1,3,5,6,9,10],[2,4,7,8,11,12]] => 10100100010 => 01000100101
[1,0,1,0,1,1,0,1,0,0,1,0] => [[1,3,5,6,8,11],[2,4,7,9,10,12]] => 10100101001 => 10010100101
[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => 10100101010 => 01010100101
[1,0,1,0,1,1,0,1,1,0,0,0] => [[1,3,5,6,8,9],[2,4,7,10,11,12]] => 10100100100 => 00100100101
[1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,6,7,11],[2,4,8,9,10,12]] => 10100010001 => 10001000101
[1,0,1,0,1,1,1,0,0,1,0,0] => [[1,3,5,6,7,10],[2,4,8,9,11,12]] => 10100010010 => 01001000101
[1,0,1,0,1,1,1,0,1,0,0,0] => [[1,3,5,6,7,9],[2,4,8,10,11,12]] => 10100010100 => 00101000101
[1,0,1,0,1,1,1,1,0,0,0,0] => [[1,3,5,6,7,8],[2,4,9,10,11,12]] => 10100001000 => 00010000101
[1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => 10010010101 => 10101001001
[1,0,1,1,0,0,1,0,1,1,0,0] => [[1,3,4,7,9,10],[2,5,6,8,11,12]] => 10010010010 => 01001001001
[1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,4,7,8,11],[2,5,6,9,10,12]] => 10010001001 => 10010001001
[1,0,1,1,0,0,1,1,0,1,0,0] => [[1,3,4,7,8,10],[2,5,6,9,11,12]] => 10010001010 => 01010001001
[1,0,1,1,0,0,1,1,1,0,0,0] => [[1,3,4,7,8,9],[2,5,6,10,11,12]] => 10010000100 => 00100001001
[1,0,1,1,0,1,0,0,1,0,1,0] => [[1,3,4,6,9,11],[2,5,7,8,10,12]] => 10010100101 => 10100101001
[1,0,1,1,0,1,0,0,1,1,0,0] => [[1,3,4,6,9,10],[2,5,7,8,11,12]] => 10010100010 => 01000101001
[1,0,1,1,0,1,0,1,0,0,1,0] => [[1,3,4,6,8,11],[2,5,7,9,10,12]] => 10010101001 => 10010101001
[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => 10010101010 => 01010101001
[1,0,1,1,0,1,0,1,1,0,0,0] => [[1,3,4,6,8,9],[2,5,7,10,11,12]] => 10010100100 => 00100101001
[1,0,1,1,0,1,1,0,0,0,1,0] => [[1,3,4,6,7,11],[2,5,8,9,10,12]] => 10010010001 => 10001001001
[1,0,1,1,0,1,1,0,0,1,0,0] => [[1,3,4,6,7,10],[2,5,8,9,11,12]] => 10010010010 => 01001001001
[1,0,1,1,0,1,1,0,1,0,0,0] => [[1,3,4,6,7,9],[2,5,8,10,11,12]] => 10010010100 => 00101001001
[1,0,1,1,0,1,1,1,0,0,0,0] => [[1,3,4,6,7,8],[2,5,9,10,11,12]] => 10010001000 => 00010001001
[1,0,1,1,1,0,0,0,1,0,1,0] => [[1,3,4,5,9,11],[2,6,7,8,10,12]] => 10001000101 => 10100010001
[1,0,1,1,1,0,0,0,1,1,0,0] => [[1,3,4,5,9,10],[2,6,7,8,11,12]] => 10001000010 => 01000010001
[1,0,1,1,1,0,0,1,0,0,1,0] => [[1,3,4,5,8,11],[2,6,7,9,10,12]] => 10001001001 => 10010010001
[1,0,1,1,1,0,0,1,0,1,0,0] => [[1,3,4,5,8,10],[2,6,7,9,11,12]] => 10001001010 => 01010010001
[1,0,1,1,1,0,0,1,1,0,0,0] => [[1,3,4,5,8,9],[2,6,7,10,11,12]] => 10001000100 => 00100010001
[1,0,1,1,1,0,1,0,0,0,1,0] => [[1,3,4,5,7,11],[2,6,8,9,10,12]] => 10001010001 => 10001010001
[1,0,1,1,1,0,1,0,0,1,0,0] => [[1,3,4,5,7,10],[2,6,8,9,11,12]] => 10001010010 => 01001010001
[1,0,1,1,1,0,1,0,1,0,0,0] => [[1,3,4,5,7,9],[2,6,8,10,11,12]] => 10001010100 => 00101010001
[1,0,1,1,1,0,1,1,0,0,0,0] => [[1,3,4,5,7,8],[2,6,9,10,11,12]] => 10001001000 => 00010010001
>>> Load all 197 entries. <<<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.
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
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.
For a standard Young tableau of size $n$ we set $w_i=1$ if $i+1$ is in a lower row than $i$, and $0$ otherwise, for $1\leq i < n$.
Map
reverse
Description
Return the reversal of a binary word.
searching the database
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