Identifier
Values
0 => 1 => [1,1] => [[1,1],[]] => 2
1 => 1 => [1,1] => [[1,1],[]] => 2
00 => 01 => [2,1] => [[2,2],[1]] => 3
01 => 10 => [1,2] => [[2,1],[]] => 3
10 => 11 => [1,1,1] => [[1,1,1],[]] => 2
11 => 11 => [1,1,1] => [[1,1,1],[]] => 2
000 => 001 => [3,1] => [[3,3],[2]] => 3
001 => 010 => [2,2] => [[3,2],[1]] => 4
010 => 101 => [1,2,1] => [[2,2,1],[1]] => 4
011 => 101 => [1,2,1] => [[2,2,1],[1]] => 4
100 => 101 => [1,2,1] => [[2,2,1],[1]] => 4
101 => 110 => [1,1,2] => [[2,1,1],[]] => 3
110 => 111 => [1,1,1,1] => [[1,1,1,1],[]] => 2
111 => 111 => [1,1,1,1] => [[1,1,1,1],[]] => 2
0000 => 0001 => [4,1] => [[4,4],[3]] => 3
0001 => 0010 => [3,2] => [[4,3],[2]] => 4
0010 => 0101 => [2,2,1] => [[3,3,2],[2,1]] => 5
0011 => 0101 => [2,2,1] => [[3,3,2],[2,1]] => 5
0100 => 1001 => [1,3,1] => [[3,3,1],[2]] => 4
0101 => 1010 => [1,2,2] => [[3,2,1],[1]] => 5
0110 => 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
0111 => 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 4
1000 => 1001 => [1,3,1] => [[3,3,1],[2]] => 4
1001 => 1010 => [1,2,2] => [[3,2,1],[1]] => 5
1010 => 1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 4
1011 => 1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 4
1100 => 1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 4
1101 => 1110 => [1,1,1,2] => [[2,1,1,1],[]] => 3
1110 => 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
1111 => 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 2
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Description
The number of corners of a skew partition.
This is also known as the number of removable cells of the skew partition.
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.
Map
valleys-to-peaks
Description
Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.