Processing math: 100%

Identifier
Values
[1,0] => [[1],[2]] => [1,1] => 11 => 1
[1,0,1,0] => [[1,3],[2,4]] => [1,2,1] => 1101 => 1
[1,1,0,0] => [[1,2],[3,4]] => [2,2] => 1010 => 1
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [1,2,2,1] => 110101 => 1
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [1,3,2] => 110010 => 1
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [2,3,1] => 101001 => 1
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [2,2,2] => 101010 => 1
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [3,3] => 100100 => 1
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [1,2,2,2,1] => 11010101 => 1
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [1,2,3,2] => 11010010 => 1
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [1,3,3,1] => 11001001 => 1
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [1,3,2,2] => 11001010 => 1
[1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [1,4,3] => 11000100 => 1
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [2,3,2,1] => 10100101 => 1
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [2,4,2] => 10100010 => 1
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [2,2,3,1] => 10101001 => 1
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [2,2,2,2] => 10101010 => 1
[1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => [2,3,3] => 10100100 => 1
[1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [3,4,1] => 10010001 => 1
[1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => [3,3,2] => 10010010 => 1
[1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => [3,2,3] => 10010100 => 1
[1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [4,4] => 10001000 => 1
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [1,2,2,3,2] => 1101010010 => 1
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [1,2,3,3,1] => 1101001001 => 1
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [1,2,3,2,2] => 1101001010 => 1
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [1,3,3,2,1] => 1100100101 => 1
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [1,3,2,3,1] => 1100101001 => 1
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => [1,3,2,2,2] => 1100101010 => 1
[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => [1,4,3,2] => 1100010010 => 1
[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => [1,5,4] => 1100001000 => 1
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [2,3,2,2,1] => 1010010101 => 1
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => [2,2,3,2,1] => 1010100101 => 1
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => [2,2,2,3,1] => 1010101001 => 1
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => [2,2,2,2,2] => 1010101010 => 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]] => [1,2,2,2,3,2] => 110101010010 => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => [1,2,2,3,3,1] => 110101001001 => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => [1,2,2,3,2,2] => 110101001010 => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => [1,2,3,3,2,1] => 110100100101 => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [[1,3,5,6,8,11],[2,4,7,9,10,12]] => [1,2,3,2,3,1] => 110100101001 => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => [1,2,3,2,2,2] => 110100101010 => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => [1,3,3,2,2,1] => 110010010101 => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [[1,3,4,6,9,11],[2,5,7,8,10,12]] => [1,3,2,3,2,1] => 110010100101 => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [[1,3,4,6,8,11],[2,5,7,9,10,12]] => [1,3,2,2,3,1] => 110010101001 => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => [1,3,2,2,2,2] => 110010101010 => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11],[3,4,6,8,10,12]] => [2,3,2,2,2,1] => 101001010101 => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [[1,2,4,7,9,11],[3,5,6,8,10,12]] => [2,2,3,2,2,1] => 101010010101 => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [[1,2,4,6,9,11],[3,5,7,8,10,12]] => [2,2,2,3,2,1] => 101010100101 => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,8,11],[3,5,7,9,10,12]] => [2,2,2,2,3,1] => 101010101001 => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8,10],[3,5,7,9,11,12]] => [2,2,2,2,2,2] => 101010101010 => 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
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of {1,,n,n+1} that contains n+1, this is the minimal element of the set.
Map
to binary word
Description
Return the composition as a binary word, treating ones as separators.
Encoding a positive integer i as the word 100 consisting of a one followed by i1 zeros, the binary word of a composition (i1,,ik) is the concatenation of of words for i1,,ik.
The image of this map contains precisely the words which do not begin with a 0.
Map
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau T the composition (c1,,ck), such that k is minimal and the numbers c1++ci+1,,c1++ci+1 form a horizontal strip in T for all 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.