Identifier
Values
[1] => 10 => [1,2] => [[2,1],[]] => 1
[2] => 100 => [1,3] => [[3,1],[]] => 1
[1,1] => 110 => [1,1,2] => [[2,1,1],[]] => 1
[3] => 1000 => [1,4] => [[4,1],[]] => 1
[2,1] => 1010 => [1,2,2] => [[3,2,1],[1]] => 2
[1,1,1] => 1110 => [1,1,1,2] => [[2,1,1,1],[]] => 1
[2,2] => 1100 => [1,1,3] => [[3,1,1],[]] => 1
[] => => [1] => [[1],[]] => 1
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searching the database for statistics with the same generating function
Description
The number of inner corners of a skew partition.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.