Identifier
Values
[1] => [1] => [1,0] => 10 => 0
[2] => [1,1] => [1,1,0,0] => 1100 => 0
[1,1] => [2] => [1,0,1,0] => 1010 => 0
[3] => [3] => [1,0,1,0,1,0] => 101010 => 0
[2,1] => [1,1,1] => [1,1,0,1,0,0] => 110100 => 0
[1,1,1] => [2,1] => [1,0,1,1,0,0] => 101100 => 0
[4] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => 0
[3,1] => [3,1] => [1,0,1,0,1,1,0,0] => 10101100 => 0
[2,2] => [4] => [1,0,1,0,1,0,1,0] => 10101010 => 0
[2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => 0
[1,1,1,1] => [2,2] => [1,1,1,0,0,0] => 111000 => 0
[3,1,1] => [3,2] => [1,0,1,1,1,0,0,0] => 10111000 => 0
[1,1,1,1,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => 0
[6] => [3,3] => [1,1,1,0,1,0,0,0] => 11101000 => 0
[1,1,1,1,1,1] => [2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => 0
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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
Glaisher-Franklin inverse
Description
The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.