Identifier
Values
0 => [2] => [[2],[]] => [2] => 1
1 => [1,1] => [[1,1],[]] => [1,1] => -1
00 => [3] => [[3],[]] => [3] => 2
01 => [2,1] => [[2,2],[1]] => [2,2] => 0
10 => [1,2] => [[2,1],[]] => [2,1] => 0
11 => [1,1,1] => [[1,1,1],[]] => [1,1,1] => -2
000 => [4] => [[4],[]] => [4] => 3
001 => [3,1] => [[3,3],[2]] => [3,3] => 1
010 => [2,2] => [[3,2],[1]] => [3,2] => 1
011 => [2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => -1
100 => [1,3] => [[3,1],[]] => [3,1] => 1
101 => [1,2,1] => [[2,2,1],[1]] => [2,2,1] => -1
110 => [1,1,2] => [[2,1,1],[]] => [2,1,1] => -1
111 => [1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1] => -3
0000 => [5] => [[5],[]] => [5] => 4
0001 => [4,1] => [[4,4],[3]] => [4,4] => 2
0010 => [3,2] => [[4,3],[2]] => [4,3] => 2
0011 => [3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => 0
0100 => [2,3] => [[4,2],[1]] => [4,2] => 2
0101 => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 0
0110 => [2,1,2] => [[3,2,2],[1,1]] => [3,2,2] => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => -2
1000 => [1,4] => [[4,1],[]] => [4,1] => 2
1001 => [1,3,1] => [[3,3,1],[2]] => [3,3,1] => 0
1010 => [1,2,2] => [[3,2,1],[1]] => [3,2,1] => 0
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] => -2
1110 => [1,1,1,2] => [[2,1,1,1],[]] => [2,1,1,1] => -2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => -4
00000 => [6] => [[6],[]] => [6] => 5
00001 => [5,1] => [[5,5],[4]] => [5,5] => 3
00010 => [4,2] => [[5,4],[3]] => [5,4] => 3
00100 => [3,3] => [[5,3],[2]] => [5,3] => 3
00110 => [3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => 1
01000 => [2,4] => [[5,2],[1]] => [5,2] => 3
01001 => [2,3,1] => [[4,4,2],[3,1]] => [4,4,2] => 1
01010 => [2,2,2] => [[4,3,2],[2,1]] => [4,3,2] => 1
01100 => [2,1,3] => [[4,2,2],[1,1]] => [4,2,2] => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [3,3,2,2] => -1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [3,2,2,2] => -1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => -3
10000 => [1,5] => [[5,1],[]] => [5,1] => 3
10001 => [1,4,1] => [[4,4,1],[3]] => [4,4,1] => 1
10010 => [1,3,2] => [[4,3,1],[2]] => [4,3,1] => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => -1
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] => -1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => [3,2,2,1] => -1
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] => 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => [3,3,1,1] => -1
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] => -3
11100 => [1,1,1,3] => [[3,1,1,1],[]] => [3,1,1,1] => -1
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => -3
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => -3
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => -5
000000 => [7] => [[7],[]] => [7] => 6
000100 => [4,3] => [[6,4],[3]] => [6,4] => 4
001000 => [3,4] => [[6,3],[2]] => [6,3] => 4
010000 => [2,5] => [[6,2],[1]] => [6,2] => 4
010100 => [2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => 2
011000 => [2,1,4] => [[5,2,2],[1,1]] => [5,2,2] => 2
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [4,2,2,2] => 0
100000 => [1,6] => [[6,1],[]] => [6,1] => 4
100010 => [1,4,2] => [[5,4,1],[3]] => [5,4,1] => 2
100100 => [1,3,3] => [[5,3,1],[2]] => [5,3,1] => 2
101000 => [1,2,4] => [[5,2,1],[1]] => [5,2,1] => 2
101010 => [1,2,2,2] => [[4,3,2,1],[2,1]] => [4,3,2,1] => 0
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]] => [4,2,2,1] => 0
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [3,2,2,2,1] => -2
110000 => [1,1,5] => [[5,1,1],[]] => [5,1,1] => 2
110001 => [1,1,4,1] => [[4,4,1,1],[3]] => [4,4,1,1] => 0
110010 => [1,1,3,2] => [[4,3,1,1],[2]] => [4,3,1,1] => 0
110100 => [1,1,2,3] => [[4,2,1,1],[1]] => [4,2,1,1] => 0
110101 => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [3,3,2,1,1] => -2
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [3,2,2,1,1] => -2
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [2,2,2,2,1,1] => -4
111000 => [1,1,1,4] => [[4,1,1,1],[]] => [4,1,1,1] => 0
111001 => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => [3,3,1,1,1] => -2
111010 => [1,1,1,2,2] => [[3,2,1,1,1],[1]] => [3,2,1,1,1] => -2
111011 => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [2,2,2,1,1,1] => -4
111100 => [1,1,1,1,3] => [[3,1,1,1,1],[]] => [3,1,1,1,1] => -2
111101 => [1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => [2,2,1,1,1,1] => -4
111110 => [1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => -4
111111 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => -6
0000000 => [8] => [[8],[]] => [8] => 7
1000000 => [1,7] => [[7,1],[]] => [7,1] => 5
1100000 => [1,1,6] => [[6,1,1],[]] => [6,1,1] => 3
1100100 => [1,1,3,3] => [[5,3,1,1],[2]] => [5,3,1,1] => 1
1110000 => [1,1,1,5] => [[5,1,1,1],[]] => [5,1,1,1] => 1
1110010 => [1,1,1,3,2] => [[4,3,1,1,1],[2]] => [4,3,1,1,1] => -1
1110110 => [1,1,1,2,1,2] => [[3,2,2,1,1,1],[1,1]] => [3,2,2,1,1,1] => -3
1111000 => [1,1,1,1,4] => [[4,1,1,1,1],[]] => [4,1,1,1,1] => -1
1111010 => [1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]] => [3,2,1,1,1,1] => -3
1111011 => [1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]] => [2,2,2,1,1,1,1] => -5
1111100 => [1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => [3,1,1,1,1,1] => -3
1111101 => [1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]] => [2,2,1,1,1,1,1] => -5
1111110 => [1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1] => -5
1111111 => [1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1] => -7
>>> Load all 121 entries. <<<
00000000 => [9] => [[9],[]] => [9] => 8
10000000 => [1,8] => [[8,1],[]] => [8,1] => 6
11000000 => [1,1,7] => [[7,1,1],[]] => [7,1,1] => 4
11100000 => [1,1,1,6] => [[6,1,1,1],[]] => [6,1,1,1] => 2
11110000 => [1,1,1,1,5] => [[5,1,1,1,1],[]] => [5,1,1,1,1] => 0
11111000 => [1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => [4,1,1,1,1,1] => -2
11111100 => [1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1] => -4
11111110 => [1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1] => -6
11111111 => [1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1] => -8
000000000 => [10] => [[10],[]] => [10] => 9
100000000 => [1,9] => [[9,1],[]] => [9,1] => 7
110000000 => [1,1,8] => [[8,1,1],[]] => [8,1,1] => 5
111000000 => [1,1,1,7] => [[7,1,1,1],[]] => [7,1,1,1] => 3
111100000 => [1,1,1,1,6] => [[6,1,1,1,1],[]] => [6,1,1,1,1] => 1
111110000 => [1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => [5,1,1,1,1,1] => -1
111111000 => [1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => [4,1,1,1,1,1,1] => -3
111111100 => [1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1,1] => -5
111111110 => [1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1,1] => -7
111111111 => [1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1,1] => -9
=> [1] => [[1],[]] => [1] => 0
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Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
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
outer shape
Description
The outer shape 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.