Identifier
Values
0 => [2] => [[2],[]] => 0
1 => [1,1] => [[1,1],[]] => 0
00 => [3] => [[3],[]] => 0
01 => [2,1] => [[2,2],[1]] => 1
10 => [1,2] => [[2,1],[]] => 0
11 => [1,1,1] => [[1,1,1],[]] => 0
000 => [4] => [[4],[]] => 0
001 => [3,1] => [[3,3],[2]] => 2
010 => [2,2] => [[3,2],[1]] => 1
011 => [2,1,1] => [[2,2,2],[1,1]] => 2
100 => [1,3] => [[3,1],[]] => 0
101 => [1,2,1] => [[2,2,1],[1]] => 1
110 => [1,1,2] => [[2,1,1],[]] => 0
111 => [1,1,1,1] => [[1,1,1,1],[]] => 0
0000 => [5] => [[5],[]] => 0
0001 => [4,1] => [[4,4],[3]] => 3
0010 => [3,2] => [[4,3],[2]] => 2
0011 => [3,1,1] => [[3,3,3],[2,2]] => 4
0100 => [2,3] => [[4,2],[1]] => 1
0101 => [2,2,1] => [[3,3,2],[2,1]] => 3
0110 => [2,1,2] => [[3,2,2],[1,1]] => 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 3
1000 => [1,4] => [[4,1],[]] => 0
1001 => [1,3,1] => [[3,3,1],[2]] => 2
1010 => [1,2,2] => [[3,2,1],[1]] => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 2
1100 => [1,1,3] => [[3,1,1],[]] => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]] => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 0
=> [1] => [[1],[]] => 0
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Description
The number of missing boxes of a skew partition.
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
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.