Identifier
Values
{{1}} => [1] => [1,0,1,0] => 1010 => 0
{{1,2}} => [2] => [1,1,0,0,1,0] => 110010 => 0
{{1},{2}} => [1,1] => [1,0,1,1,0,0] => 101100 => 0
{{1,2,3}} => [3] => [1,1,1,0,0,0,1,0] => 11100010 => 0
{{1,2},{3}} => [2,1] => [1,0,1,0,1,0] => 101010 => 0
{{1,3},{2}} => [2,1] => [1,0,1,0,1,0] => 101010 => 0
{{1},{2,3}} => [2,1] => [1,0,1,0,1,0] => 101010 => 0
{{1},{2},{3}} => [1,1,1] => [1,0,1,1,1,0,0,0] => 10111000 => 0
{{1,2,3},{4}} => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
{{1,2,4},{3}} => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
{{1,2},{3,4}} => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
{{1,2},{3},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
{{1,3,4},{2}} => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
{{1,3},{2,4}} => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
{{1,3},{2},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
{{1,4},{2,3}} => [2,2] => [1,1,0,0,1,1,0,0] => 11001100 => 0
{{1},{2,3,4}} => [3,1] => [1,1,0,1,0,0,1,0] => 11010010 => 0
{{1},{2,3},{4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
{{1,4},{2},{3}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
{{1},{2,4},{3}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
{{1},{2},{3,4}} => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
{{1,2,3},{4,5}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,2,3},{4},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,2,4},{3,5}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,2,4},{3},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,2,5},{3,4}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,2},{3,4,5}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,2},{3,4},{5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,2,5},{3},{4}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,2},{3,5},{4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,2},{3},{4,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,3,4},{2,5}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,3,4},{2},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,3,5},{2,4}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,3},{2,4,5}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,3},{2,4},{5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,3,5},{2},{4}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,3},{2,5},{4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,3},{2},{4,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,4,5},{2,3}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,4},{2,3,5}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1,4},{2,3},{5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,5},{2,3,4}} => [3,2] => [1,1,0,0,1,0,1,0] => 11001010 => 0
{{1},{2,3,4},{5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,5},{2,3},{4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1},{2,3,5},{4}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1},{2,3},{4,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,4,5},{2},{3}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,4},{2,5},{3}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,4},{2},{3,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,5},{2,4},{3}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1},{2,4,5},{3}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1},{2,4},{3,5}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1,5},{2},{3,4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1},{2,5},{3,4}} => [2,2,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
{{1},{2},{3,4,5}} => [3,1,1] => [1,0,1,1,0,0,1,0] => 10110010 => 0
{{1,2,3},{4,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,3},{4,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,3},{4},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,4},{3,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,4},{3,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,4},{3},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,5},{3,4},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2},{3,4,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,6},{3,4},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2},{3,4,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,5},{3,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,5},{3},{4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,6},{3,5},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2},{3,5,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2,6},{3},{4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,2},{3},{4,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,4},{2,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,4},{2,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,4},{2},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,5},{2,4},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3},{2,4,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,6},{2,4},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3},{2,4,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,5},{2,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,5},{2},{4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,6},{2,5},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3},{2,5,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3,6},{2},{4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,3},{2},{4,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4,5},{2,3},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4},{2,3,5},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4,6},{2,3},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4},{2,3,6},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5},{2,3,4},{6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,6},{2,3,4},{5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,3,4},{5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5,6},{2,3},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5},{2,3,6},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,6},{2,3,5},{4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,3,5},{4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,3,6},{4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,3},{4,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4,5},{2,6},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4,5},{2},{3,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4,6},{2,5},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
>>> Load all 116 entries. <<<
{{1,4},{2,5,6},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4,6},{2},{3,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,4},{2},{3,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5,6},{2,4},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5},{2,4,6},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,6},{2,4,5},{3}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,4,5},{3,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,4,6},{3,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,4},{3,5,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5,6},{2},{3,4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,5},{2},{3,4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,5,6},{3,4}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,5},{3,4,6}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1,6},{2},{3,4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
{{1},{2,6},{3,4,5}} => [3,2,1] => [1,0,1,0,1,0,1,0] => 10101010 => 0
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 number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
shape
Description
Sends a set partition to the integer partition obtained by the sizes of the blocks.