Processing math: 100%

Identifier
Values
[1,0] => [[1],[2]] => 1 => 1 => 1
[1,0,1,0] => [[1,3],[2,4]] => 101 => 101 => 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 => 01101 => 3
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 10010 => 00110 => 2
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 01001 => 01001 => 2
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 01010 => 01100 => 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 => 1001101 => 4
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 1010010 => 0100110 => 3
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1001001 => 1000101 => 3
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 1001010 => 1000110 => 3
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 1000100 => 1000010 => 2
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 0100101 => 0011001 => 3
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 0100010 => 0001100 => 2
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 0101001 => 1001001 => 3
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 0101010 => 1001100 => 3
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 0100100 => 1000100 => 2
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 0010001 => 0010001 => 2
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 0010010 => 0011000 => 2
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 0010100 => 1001000 => 2
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 0001000 => 0010000 => 1
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => 101010010 => 100100110 => 4
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => 100100101 => 010001101 => 4
[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => 100010001 => 010000101 => 3
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => 100010010 => 010000110 => 3
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => 100001000 => 010000010 => 2
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => 010001010 => 100001100 => 3
[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => 010000100 => 100000100 => 2
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => 010101001 => 011001001 => 4
[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => 010010001 => 010001001 => 3
[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => 010001000 => 010000100 => 2
[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => 001001001 => 100010001 => 3
[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => 001000100 => 100001000 => 2
[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => 001010001 => 010010001 => 3
[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => 001001000 => 010001000 => 2
[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => 000100001 => 000100001 => 2
[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => 000100100 => 100010000 => 2
[1,1,1,1,0,1,0,0,0,0] => [[1,2,3,4,6],[5,7,8,9,10]] => 000101000 => 010010000 => 2
[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => 000010000 => 000100000 => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00096Foata bijection.
Map
descent word
Description
The descent word of a standard Young tableau.
For a standard Young tableau of size n we set wi=1 if i+1 is in a lower row than i, and 0 otherwise, for 1i<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.