Identifier
-
Mp00180:
Integer compositions
—to ribbon⟶
Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤ
Values
[2,1] => [[2,2],[1]] => [1] => 1 => 1
[1,2,1] => [[2,2,1],[1]] => [1] => 1 => 1
[2,1,1] => [[2,2,2],[1,1]] => [1,1] => 11 => 2
[2,2] => [[3,2],[1]] => [1] => 1 => 1
[3,1] => [[3,3],[2]] => [2] => 0 => 0
[1,1,2,1] => [[2,2,1,1],[1]] => [1] => 1 => 1
[1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 11 => 2
[1,2,2] => [[3,2,1],[1]] => [1] => 1 => 1
[1,3,1] => [[3,3,1],[2]] => [2] => 0 => 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 111 => 3
[2,1,2] => [[3,2,2],[1,1]] => [1,1] => 11 => 2
[2,2,1] => [[3,3,2],[2,1]] => [2,1] => 01 => 1
[2,3] => [[4,2],[1]] => [1] => 1 => 1
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => 00 => 0
[3,2] => [[4,3],[2]] => [2] => 0 => 0
[4,1] => [[4,4],[3]] => [3] => 1 => 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 1 => 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 11 => 2
[1,1,2,2] => [[3,2,1,1],[1]] => [1] => 1 => 1
[1,1,3,1] => [[3,3,1,1],[2]] => [2] => 0 => 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 111 => 3
[1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 11 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 01 => 1
[1,2,3] => [[4,2,1],[1]] => [1] => 1 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 00 => 0
[1,3,2] => [[4,3,1],[2]] => [2] => 0 => 0
[1,4,1] => [[4,4,1],[3]] => [3] => 1 => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1111 => 4
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 111 => 3
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 011 => 2
[2,1,3] => [[4,2,2],[1,1]] => [1,1] => 11 => 2
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 001 => 1
[2,2,2] => [[4,3,2],[2,1]] => [2,1] => 01 => 1
[2,3,1] => [[4,4,2],[3,1]] => [3,1] => 11 => 2
[2,4] => [[5,2],[1]] => [1] => 1 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 000 => 0
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => 00 => 0
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => 10 => 1
[3,3] => [[5,3],[2]] => [2] => 0 => 0
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => 11 => 2
[4,2] => [[5,4],[3]] => [3] => 1 => 1
[5,1] => [[5,5],[4]] => [4] => 0 => 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [1] => 1 => 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 11 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => [1] => 1 => 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 0 => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 111 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 11 => 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 01 => 1
[1,1,2,3] => [[4,2,1,1],[1]] => [1] => 1 => 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 00 => 0
[1,1,3,2] => [[4,3,1,1],[2]] => [2] => 0 => 0
[1,1,4,1] => [[4,4,1,1],[3]] => [3] => 1 => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 1111 => 4
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 111 => 3
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 011 => 2
[1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 11 => 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 001 => 1
[1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => 01 => 1
[1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 11 => 2
[1,2,4] => [[5,2,1],[1]] => [1] => 1 => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 000 => 0
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 00 => 0
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 10 => 1
[1,3,3] => [[5,3,1],[2]] => [2] => 0 => 0
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 11 => 2
[1,4,2] => [[5,4,1],[3]] => [3] => 1 => 1
[1,5,1] => [[5,5,1],[4]] => [4] => 0 => 0
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 11111 => 5
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1111 => 4
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 0111 => 3
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 111 => 3
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 0011 => 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 011 => 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 111 => 3
[2,1,4] => [[5,2,2],[1,1]] => [1,1] => 11 => 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 0001 => 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 001 => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 101 => 2
[2,2,3] => [[5,3,2],[2,1]] => [2,1] => 01 => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 111 => 3
[2,3,2] => [[5,4,2],[3,1]] => [3,1] => 11 => 2
[2,4,1] => [[5,5,2],[4,1]] => [4,1] => 01 => 1
[2,5] => [[6,2],[1]] => [1] => 1 => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 0000 => 0
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 000 => 0
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 100 => 1
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => 00 => 0
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 110 => 2
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => 10 => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => 00 => 0
[3,4] => [[6,3],[2]] => [2] => 0 => 0
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 111 => 3
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => 11 => 2
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => 01 => 1
[4,3] => [[6,4],[3]] => [3] => 1 => 1
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => 00 => 0
[5,2] => [[6,5],[4]] => [4] => 0 => 0
[6,1] => [[6,6],[5]] => [5] => 1 => 1
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [1] => 1 => 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [1,1] => 11 => 2
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Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
This is also known as the Hamming weight of the word.
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.
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
odd parts
Description
Return the binary word indicating which parts of the partition are odd.
Map
inner shape
Description
The inner shape of a skew partition.
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