Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤ
Values
0 => [2] => [[2],[]] => [2] => 1
1 => [1,1] => [[1,1],[]] => [1,1] => 0
00 => [3] => [[3],[]] => [3] => 0
01 => [2,1] => [[2,2],[1]] => [2,2] => 2
10 => [1,2] => [[2,1],[]] => [2,1] => 1
11 => [1,1,1] => [[1,1,1],[]] => [1,1,1] => 0
000 => [4] => [[4],[]] => [4] => 0
001 => [3,1] => [[3,3],[2]] => [3,3] => 0
010 => [2,2] => [[3,2],[1]] => [3,2] => 1
011 => [2,1,1] => [[2,2,2],[1,1]] => [2,2,2] => 3
100 => [1,3] => [[3,1],[]] => [3,1] => 0
101 => [1,2,1] => [[2,2,1],[1]] => [2,2,1] => 2
110 => [1,1,2] => [[2,1,1],[]] => [2,1,1] => 1
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] => 0
0010 => [3,2] => [[4,3],[2]] => [4,3] => 0
0011 => [3,1,1] => [[3,3,3],[2,2]] => [3,3,3] => 0
0100 => [2,3] => [[4,2],[1]] => [4,2] => 1
0101 => [2,2,1] => [[3,3,2],[2,1]] => [3,3,2] => 1
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] => 4
1000 => [1,4] => [[4,1],[]] => [4,1] => 0
1001 => [1,3,1] => [[3,3,1],[2]] => [3,3,1] => 0
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] => 3
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] => 1
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] => 0
00010 => [4,2] => [[5,4],[3]] => [5,4] => 0
00011 => [4,1,1] => [[4,4,4],[3,3]] => [4,4,4] => 0
00100 => [3,3] => [[5,3],[2]] => [5,3] => 0
00101 => [3,2,1] => [[4,4,3],[3,2]] => [4,4,3] => 0
00110 => [3,1,2] => [[4,3,3],[2,2]] => [4,3,3] => 0
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [3,3,3,3] => 0
01000 => [2,4] => [[5,2],[1]] => [5,2] => 1
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
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [3,3,3,2] => 1
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] => 2
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] => 5
10000 => [1,5] => [[5,1],[]] => [5,1] => 0
10001 => [1,4,1] => [[4,4,1],[3]] => [4,4,1] => 0
10010 => [1,3,2] => [[4,3,1],[2]] => [4,3,1] => 0
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [3,3,3,1] => 0
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] => 2
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [2,2,2,2,1] => 4
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] => 0
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] => 0
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => [2,2,1,1,1] => 2
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => 1
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] => 0
000010 => [5,2] => [[6,5],[4]] => [6,5] => 0
000011 => [5,1,1] => [[5,5,5],[4,4]] => [5,5,5] => 0
000100 => [4,3] => [[6,4],[3]] => [6,4] => 0
000101 => [4,2,1] => [[5,5,4],[4,3]] => [5,5,4] => 0
000110 => [4,1,2] => [[5,4,4],[3,3]] => [5,4,4] => 0
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [4,4,4,4] => 0
001000 => [3,4] => [[6,3],[2]] => [6,3] => 0
001001 => [3,3,1] => [[5,5,3],[4,2]] => [5,5,3] => 0
001010 => [3,2,2] => [[5,4,3],[3,2]] => [5,4,3] => 0
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [4,4,4,3] => 0
001100 => [3,1,3] => [[5,3,3],[2,2]] => [5,3,3] => 0
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [4,4,3,3] => 0
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [4,3,3,3] => 0
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [3,3,3,3,3] => 0
010000 => [2,5] => [[6,2],[1]] => [6,2] => 1
010001 => [2,4,1] => [[5,5,2],[4,1]] => [5,5,2] => 1
010010 => [2,3,2] => [[5,4,2],[3,1]] => [5,4,2] => 1
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => [4,4,4,2] => 1
010100 => [2,2,3] => [[5,3,2],[2,1]] => [5,3,2] => 1
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [4,4,3,2] => 1
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [4,3,3,2] => 1
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [3,3,3,3,2] => 1
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] => 2
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [4,3,2,2] => 2
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [3,3,3,2,2] => 2
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] => 3
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] => 6
100000 => [1,6] => [[6,1],[]] => [6,1] => 0
100001 => [1,5,1] => [[5,5,1],[4]] => [5,5,1] => 0
100010 => [1,4,2] => [[5,4,1],[3]] => [5,4,1] => 0
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => [4,4,4,1] => 0
100100 => [1,3,3] => [[5,3,1],[2]] => [5,3,1] => 0
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [4,4,3,1] => 0
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [4,3,3,1] => 0
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Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons St001484The number of singletons of an integer partition. in all partitions of $n-1$, see [1].
The total number of twos in all partitions of $n$ is equal to the total number of singletons St001484The number of singletons of an integer partition. in all partitions of $n-1$, see [1].
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
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.
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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