Identifier
-
Mp00033:
Dyck paths
—to two-row standard tableau⟶
Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000288: Binary words ⟶ ℤ
Values
[1,0] => [[1],[2]] => 1 => 1 => 1
[1,0,1,0] => [[1,3],[2,4]] => 101 => 011 => 2
[1,1,0,0] => [[1,2],[3,4]] => 010 => 100 => 1
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 10101 => 01011 => 3
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 10010 => 00101 => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 01001 => 10010 => 2
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 01010 => 10100 => 2
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 00100 => 01000 => 1
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 1010101 => 0101011 => 4
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 1010010 => 0100101 => 3
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1001001 => 0010011 => 3
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 1001010 => 0010101 => 3
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 1000100 => 0001001 => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 0100101 => 1001010 => 3
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 0100010 => 1000100 => 2
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 0101001 => 1010010 => 3
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 0101010 => 1010100 => 3
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 0100100 => 1001000 => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 0010001 => 0100010 => 2
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 0010010 => 0100100 => 2
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 0010100 => 0101000 => 2
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 0001000 => 0010000 => 1
[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => 101010101 => 010101011 => 5
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => 101010010 => 010100101 => 4
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => 101001001 => 010010011 => 4
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => 101001010 => 010010101 => 4
[1,0,1,0,1,1,1,0,0,0] => [[1,3,5,6,7],[2,4,8,9,10]] => 101000100 => 010001001 => 3
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 100100101 => 001001011 => 4
[1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => 100100010 => 001000101 => 3
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => 100101010 => 001010101 => 4
[1,0,1,1,0,1,1,0,0,0] => [[1,3,4,6,7],[2,5,8,9,10]] => 100100100 => 001001001 => 3
[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => 100010001 => 000100011 => 3
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => 100010010 => 000100101 => 3
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => 100001000 => 000010001 => 2
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => 010010101 => 100101010 => 4
[1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => 010010010 => 100100100 => 3
[1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => 010001001 => 100010010 => 3
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => 010001010 => 100010100 => 3
[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => 010000100 => 100001000 => 2
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => 010100101 => 101001010 => 4
[1,1,0,1,0,0,1,1,0,0] => [[1,2,4,7,8],[3,5,6,9,10]] => 010100010 => 101000100 => 3
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => 010101001 => 101010010 => 4
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => 010101010 => 101010100 => 4
[1,1,0,1,0,1,1,0,0,0] => [[1,2,4,6,7],[3,5,8,9,10]] => 010100100 => 101001000 => 3
[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => 010010001 => 100100010 => 3
[1,1,0,1,1,0,0,1,0,0] => [[1,2,4,5,8],[3,6,7,9,10]] => 010010010 => 100100100 => 3
[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => 010001000 => 100010000 => 2
[1,1,1,0,0,0,1,0,1,0] => [[1,2,3,7,9],[4,5,6,8,10]] => 001000101 => 010001010 => 3
[1,1,1,0,0,0,1,1,0,0] => [[1,2,3,7,8],[4,5,6,9,10]] => 001000010 => 010000100 => 2
[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => 001001001 => 010010010 => 3
[1,1,1,0,0,1,0,1,0,0] => [[1,2,3,6,8],[4,5,7,9,10]] => 001001010 => 010010100 => 3
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => 001000100 => 010001000 => 2
[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => 001010001 => 010100010 => 3
[1,1,1,0,1,0,0,1,0,0] => [[1,2,3,5,8],[4,6,7,9,10]] => 001010010 => 010100100 => 3
[1,1,1,0,1,0,1,0,0,0] => [[1,2,3,5,7],[4,6,8,9,10]] => 001010100 => 010101000 => 3
[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => 001001000 => 010010000 => 2
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => 000100001 => 001000010 => 2
[1,1,1,1,0,0,0,1,0,0] => [[1,2,3,4,8],[5,6,7,9,10]] => 000100010 => 001000100 => 2
[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => 000100100 => 001001000 => 2
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => 000100000 => 1
[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 => 01010100101 => 5
[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 => 5
[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 => 01010001001 => 4
[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 => 01001000101 => 4
[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 => 01001010101 => 5
[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 => 01001001001 => 4
[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 => 01000100101 => 4
[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 => 01000101001 => 4
[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 => 01000010001 => 3
[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 => 00100100101 => 4
[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 => 00100010101 => 4
[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 => 3
[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 => 00101000101 => 4
[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 => 00101001001 => 4
[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 => 00100100101 => 4
[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 => 00100101001 => 4
[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 => 00100010001 => 3
[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 => 00010000101 => 3
[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 => 00010001001 => 3
[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 => 3
[1,1,0,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11],[3,4,6,8,10,12]] => 01001010101 => 10010101010 => 5
[1,1,0,0,1,0,1,0,1,1,0,0] => [[1,2,5,7,9,10],[3,4,6,8,11,12]] => 01001010010 => 10010100100 => 4
[1,1,0,0,1,0,1,1,0,0,1,0] => [[1,2,5,7,8,11],[3,4,6,9,10,12]] => 01001001001 => 10010010010 => 4
[1,1,0,0,1,0,1,1,0,1,0,0] => [[1,2,5,7,8,10],[3,4,6,9,11,12]] => 01001001010 => 10010010100 => 4
[1,1,0,0,1,0,1,1,1,0,0,0] => [[1,2,5,7,8,9],[3,4,6,10,11,12]] => 01001000100 => 10010001000 => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => [[1,2,5,6,9,11],[3,4,7,8,10,12]] => 01000100101 => 10001001010 => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => [[1,2,5,6,9,10],[3,4,7,8,11,12]] => 01000100010 => 10001000100 => 3
[1,1,0,0,1,1,0,1,0,0,1,0] => [[1,2,5,6,8,11],[3,4,7,9,10,12]] => 01000101001 => 10001010010 => 4
[1,1,0,0,1,1,0,1,0,1,0,0] => [[1,2,5,6,8,10],[3,4,7,9,11,12]] => 01000101010 => 10001010100 => 4
[1,1,0,0,1,1,0,1,1,0,0,0] => [[1,2,5,6,8,9],[3,4,7,10,11,12]] => 01000100100 => 10001001000 => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [[1,2,5,6,7,11],[3,4,8,9,10,12]] => 01000010001 => 10000100010 => 3
[1,1,0,0,1,1,1,0,0,1,0,0] => [[1,2,5,6,7,10],[3,4,8,9,11,12]] => 01000010010 => 10000100100 => 3
[1,1,0,0,1,1,1,0,1,0,0,0] => [[1,2,5,6,7,9],[3,4,8,10,11,12]] => 01000010100 => 10000101000 => 3
[1,1,0,0,1,1,1,1,0,0,0,0] => [[1,2,5,6,7,8],[3,4,9,10,11,12]] => 01000001000 => 10000010000 => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [[1,2,4,7,9,11],[3,5,6,8,10,12]] => 01010010101 => 10100101010 => 5
[1,1,0,1,0,0,1,0,1,1,0,0] => [[1,2,4,7,9,10],[3,5,6,8,11,12]] => 01010010010 => 10100100100 => 4
[1,1,0,1,0,0,1,1,0,0,1,0] => [[1,2,4,7,8,11],[3,5,6,9,10,12]] => 01010001001 => 10100010010 => 4
[1,1,0,1,0,0,1,1,0,1,0,0] => [[1,2,4,7,8,10],[3,5,6,9,11,12]] => 01010001010 => 10100010100 => 4
[1,1,0,1,0,0,1,1,1,0,0,0] => [[1,2,4,7,8,9],[3,5,6,10,11,12]] => 01010000100 => 10100001000 => 3
[1,1,0,1,0,1,0,0,1,0,1,0] => [[1,2,4,6,9,11],[3,5,7,8,10,12]] => 01010100101 => 10101001010 => 5
[1,1,0,1,0,1,0,0,1,1,0,0] => [[1,2,4,6,9,10],[3,5,7,8,11,12]] => 01010100010 => 10101000100 => 4
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Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
This is also known as the Hamming weight of the word.
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
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
rotate front-to-back
Description
The rotation of a binary word, first letter last.
This is the word obtained by moving the first letter to the end.
This is the word obtained by moving the first letter to the end.
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