Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤ
Values
0 => [2] => [[2],[]] => [2] => 0
1 => [1,1] => [[1,1],[]] => [1,1] => 0
00 => [3] => [[3],[]] => [3] => 0
01 => [2,1] => [[2,2],[1]] => [2,2] => 1
10 => [1,2] => [[2,1],[]] => [2,1] => 0
11 => [1,1,1] => [[1,1,1],[]] => [1,1,1] => 0
000 => [4] => [[4],[]] => [4] => 0
001 => [3,1] => [[3,3],[2]] => [3,3] => 2
010 => [2,2] => [[3,2],[1]] => [3,2] => 1
011 => [2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => 2
100 => [1,3] => [[3,1],[]] => [3,1] => 0
101 => [1,2,1] => [[2,2,1],[1]] => [2,2,1] => 1
110 => [1,1,2] => [[2,1,1],[]] => [2,1,1] => 0
111 => [1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1] => 0
0000 => [5] => [[5],[]] => [5] => 0
0001 => [4,1] => [[4,4],[3]] => [4,4] => 3
0010 => [3,2] => [[4,3],[2]] => [4,3] => 2
0011 => [3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => 4
0100 => [2,3] => [[4,2],[1]] => [4,2] => 1
0101 => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 3
0110 => [2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => 3
1000 => [1,4] => [[4,1],[]] => [4,1] => 0
1001 => [1,3,1] => [[3,3,1],[2]] => [3,3,1] => 2
1010 => [1,2,2] => [[3,2,1],[1]] => [3,2,1] => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [2,2,2,1] => 2
1100 => [1,1,3] => [[3,1,1],[]] => [3,1,1] => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]] => [2,2,1,1] => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]] => [2,1,1,1] => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => 0
00000 => [6] => [[6],[]] => [6] => 0
00001 => [5,1] => [[5,5],[4]] => [5,5] => 4
00010 => [4,2] => [[5,4],[3]] => [5,4] => 3
00011 => [4,1,1] => [[4,4,4],[3,3]] => [4,4,4] => 6
00100 => [3,3] => [[5,3],[2]] => [5,3] => 2
00101 => [3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => 5
00110 => [3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => 4
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [3,3,3,3] => 6
01000 => [2,4] => [[5,2],[1]] => [5,2] => 1
01001 => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 4
01010 => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 3
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [3,3,3,2] => 5
01100 => [2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => 2
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [3,3,2,2] => 4
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => 3
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => 4
10000 => [1,5] => [[5,1],[]] => [5,1] => 0
10001 => [1,4,1] => [[4,4,1],[3]] => [4,4,1] => 3
10010 => [1,3,2] => [[4,3,1],[2]] => [4,3,1] => 2
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => 4
10100 => [1,2,3] => [[4,2,1],[1]] => [4,2,1] => 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => [3,3,2,1] => 3
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => [3,2,2,1] => 2
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => 3
11000 => [1,1,4] => [[4,1,1],[]] => [4,1,1] => 0
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => [3,3,1,1] => 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]] => [3,2,1,1] => 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [2,2,2,1,1] => 2
11100 => [1,1,1,3] => [[3,1,1,1],[]] => [3,1,1,1] => 0
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => 0
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => 0
000000 => [7] => [[7],[]] => [7] => 0
000001 => [6,1] => [[6,6],[5]] => [6,6] => 5
000010 => [5,2] => [[6,5],[4]] => [6,5] => 4
000011 => [5,1,1] => [[5,5,5],[4,4]] => [5,5,5] => 8
000100 => [4,3] => [[6,4],[3]] => [6,4] => 3
000101 => [4,2,1] => [[5,5,4],[4,3]] => [5,5,4] => 7
000110 => [4,1,2] => [[5,4,4],[3,3]] => [5,4,4] => 6
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [4,4,4,4] => 9
001000 => [3,4] => [[6,3],[2]] => [6,3] => 2
001001 => [3,3,1] => [[5,5,3],[4,2]] => [5,5,3] => 6
001010 => [3,2,2] => [[5,4,3],[3,2]] => [5,4,3] => 5
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [4,4,4,3] => 8
001100 => [3,1,3] => [[5,3,3],[2,2]] => [5,3,3] => 4
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [4,4,3,3] => 7
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [4,3,3,3] => 6
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [3,3,3,3,3] => 8
010000 => [2,5] => [[6,2],[1]] => [6,2] => 1
010001 => [2,4,1] => [[5,5,2],[4,1]] => [5,5,2] => 5
010010 => [2,3,2] => [[5,4,2],[3,1]] => [5,4,2] => 4
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => [4,4,4,2] => 7
010100 => [2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => 3
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [4,4,3,2] => 6
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [4,3,3,2] => 5
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [3,3,3,3,2] => 7
011000 => [2,1,4] => [[5,2,2],[1,1]] => [5,2,2] => 2
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [4,4,2,2] => 5
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => 4
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [3,3,3,2,2] => 6
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => 3
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [3,3,2,2,2] => 5
011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [3,2,2,2,2] => 4
011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2] => 5
100000 => [1,6] => [[6,1],[]] => [6,1] => 0
100001 => [1,5,1] => [[5,5,1],[4]] => [5,5,1] => 4
100010 => [1,4,2] => [[5,4,1],[3]] => [5,4,1] => 3
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => [4,4,4,1] => 6
100100 => [1,3,3] => [[5,3,1],[2]] => [5,3,1] => 2
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => 5
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => 4
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Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Map
outer shape
Description
The outer shape of the 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.
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.
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.
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