Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001595: Skew partitions ⟶ ℤ
Values
0 => [2] => [[2],[]] => 1
1 => [1,1] => [[1,1],[]] => 1
00 => [3] => [[3],[]] => 1
01 => [2,1] => [[2,2],[1]] => 2
10 => [1,2] => [[2,1],[]] => 2
11 => [1,1,1] => [[1,1,1],[]] => 1
000 => [4] => [[4],[]] => 1
001 => [3,1] => [[3,3],[2]] => 3
010 => [2,2] => [[3,2],[1]] => 5
011 => [2,1,1] => [[2,2,2],[1,1]] => 3
100 => [1,3] => [[3,1],[]] => 3
101 => [1,2,1] => [[2,2,1],[1]] => 5
110 => [1,1,2] => [[2,1,1],[]] => 3
111 => [1,1,1,1] => [[1,1,1,1],[]] => 1
0000 => [5] => [[5],[]] => 1
0001 => [4,1] => [[4,4],[3]] => 4
0010 => [3,2] => [[4,3],[2]] => 9
0011 => [3,1,1] => [[3,3,3],[2,2]] => 6
0100 => [2,3] => [[4,2],[1]] => 9
0101 => [2,2,1] => [[3,3,2],[2,1]] => 16
0110 => [2,1,2] => [[3,2,2],[1,1]] => 11
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => 4
1000 => [1,4] => [[4,1],[]] => 4
1001 => [1,3,1] => [[3,3,1],[2]] => 11
1010 => [1,2,2] => [[3,2,1],[1]] => 16
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => 9
1100 => [1,1,3] => [[3,1,1],[]] => 6
1101 => [1,1,2,1] => [[2,2,1,1],[1]] => 9
1110 => [1,1,1,2] => [[2,1,1,1],[]] => 4
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => 1
00000 => [6] => [[6],[]] => 1
00001 => [5,1] => [[5,5],[4]] => 5
00010 => [4,2] => [[5,4],[3]] => 14
00011 => [4,1,1] => [[4,4,4],[3,3]] => 10
00100 => [3,3] => [[5,3],[2]] => 19
00101 => [3,2,1] => [[4,4,3],[3,2]] => 35
00110 => [3,1,2] => [[4,3,3],[2,2]] => 26
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => 10
01000 => [2,4] => [[5,2],[1]] => 14
01001 => [2,3,1] => [[4,4,2],[3,1]] => 40
01010 => [2,2,2] => [[4,3,2],[2,1]] => 61
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => 35
01100 => [2,1,3] => [[4,2,2],[1,1]] => 26
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => 40
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => 19
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => 5
10000 => [1,5] => [[5,1],[]] => 5
10001 => [1,4,1] => [[4,4,1],[3]] => 19
10010 => [1,3,2] => [[4,3,1],[2]] => 40
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => 26
10100 => [1,2,3] => [[4,2,1],[1]] => 35
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => 61
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => 40
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => 14
11000 => [1,1,4] => [[4,1,1],[]] => 10
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => 26
11010 => [1,1,2,2] => [[3,2,1,1],[1]] => 35
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => 19
11100 => [1,1,1,3] => [[3,1,1,1],[]] => 10
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => 14
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => 5
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => 1
000000 => [7] => [[7],[]] => 1
000001 => [6,1] => [[6,6],[5]] => 6
000010 => [5,2] => [[6,5],[4]] => 20
000011 => [5,1,1] => [[5,5,5],[4,4]] => 15
000100 => [4,3] => [[6,4],[3]] => 34
000101 => [4,2,1] => [[5,5,4],[4,3]] => 64
000110 => [4,1,2] => [[5,4,4],[3,3]] => 50
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => 20
001000 => [3,4] => [[6,3],[2]] => 34
001001 => [3,3,1] => [[5,5,3],[4,2]] => 99
001010 => [3,2,2] => [[5,4,3],[3,2]] => 155
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => 90
001100 => [3,1,3] => [[5,3,3],[2,2]] => 71
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => 111
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => 55
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => 15
010000 => [2,5] => [[6,2],[1]] => 20
010001 => [2,4,1] => [[5,5,2],[4,1]] => 78
010010 => [2,3,2] => [[5,4,2],[3,1]] => 169
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => 111
010100 => [2,2,3] => [[5,3,2],[2,1]] => 155
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => 272
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => 181
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => 64
011000 => [2,1,4] => [[5,2,2],[1,1]] => 50
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => 132
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => 181
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => 99
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => 55
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => 78
011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => 29
011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => 6
100000 => [1,6] => [[6,1],[]] => 6
100001 => [1,5,1] => [[5,5,1],[4]] => 29
100010 => [1,4,2] => [[5,4,1],[3]] => 78
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => 55
100100 => [1,3,3] => [[5,3,1],[2]] => 99
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => 181
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => 132
>>> Load all 127 entries. <<<
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Description
The number of standard Young tableaux 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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