Identifier
-
Mp00178:
Binary words
—to composition⟶
Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤ (values match St000318The number of addable cells of the Ferrers diagram of an integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)
Values
0 => [2] => [[2],[]] => [] => 0
1 => [1,1] => [[1,1],[]] => [] => 0
00 => [3] => [[3],[]] => [] => 0
01 => [2,1] => [[2,2],[1]] => [1] => 1
10 => [1,2] => [[2,1],[]] => [] => 0
11 => [1,1,1] => [[1,1,1],[]] => [] => 0
000 => [4] => [[4],[]] => [] => 0
001 => [3,1] => [[3,3],[2]] => [2] => 1
010 => [2,2] => [[3,2],[1]] => [1] => 1
011 => [2,1,1] => [[2,2,2],[1,1]] => [1,1] => 1
100 => [1,3] => [[3,1],[]] => [] => 0
101 => [1,2,1] => [[2,2,1],[1]] => [1] => 1
110 => [1,1,2] => [[2,1,1],[]] => [] => 0
111 => [1,1,1,1] => [[1,1,1,1],[]] => [] => 0
0000 => [5] => [[5],[]] => [] => 0
0001 => [4,1] => [[4,4],[3]] => [3] => 1
0010 => [3,2] => [[4,3],[2]] => [2] => 1
0011 => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 1
0100 => [2,3] => [[4,2],[1]] => [1] => 1
0101 => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 2
0110 => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 1
1000 => [1,4] => [[4,1],[]] => [] => 0
1001 => [1,3,1] => [[3,3,1],[2]] => [2] => 1
1010 => [1,2,2] => [[3,2,1],[1]] => [1] => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 1
1100 => [1,1,3] => [[3,1,1],[]] => [] => 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]] => [1] => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]] => [] => 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => [] => 0
00000 => [6] => [[6],[]] => [] => 0
00001 => [5,1] => [[5,5],[4]] => [4] => 1
00010 => [4,2] => [[5,4],[3]] => [3] => 1
00011 => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 1
00100 => [3,3] => [[5,3],[2]] => [2] => 1
00101 => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 2
00110 => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 1
01000 => [2,4] => [[5,2],[1]] => [1] => 1
01001 => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 2
01010 => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 2
01100 => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 2
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 1
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
10000 => [1,5] => [[5,1],[]] => [] => 0
10001 => [1,4,1] => [[4,4,1],[3]] => [3] => 1
10010 => [1,3,2] => [[4,3,1],[2]] => [2] => 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 1
10100 => [1,2,3] => [[4,2,1],[1]] => [1] => 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 2
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 1
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 1
11000 => [1,1,4] => [[4,1,1],[]] => [] => 0
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]] => [1] => 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 1
11100 => [1,1,1,3] => [[3,1,1,1],[]] => [] => 0
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => [1] => 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => [] => 0
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [] => 0
000000 => [7] => [[7],[]] => [] => 0
000001 => [6,1] => [[6,6],[5]] => [5] => 1
000010 => [5,2] => [[6,5],[4]] => [4] => 1
000011 => [5,1,1] => [[5,5,5],[4,4]] => [4,4] => 1
000100 => [4,3] => [[6,4],[3]] => [3] => 1
000101 => [4,2,1] => [[5,5,4],[4,3]] => [4,3] => 2
000110 => [4,1,2] => [[5,4,4],[3,3]] => [3,3] => 1
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 1
001000 => [3,4] => [[6,3],[2]] => [2] => 1
001001 => [3,3,1] => [[5,5,3],[4,2]] => [4,2] => 2
001010 => [3,2,2] => [[5,4,3],[3,2]] => [3,2] => 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => 2
001100 => [3,1,3] => [[5,3,3],[2,2]] => [2,2] => 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => 2
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => 1
001111 => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 1
010000 => [2,5] => [[6,2],[1]] => [1] => 1
010001 => [2,4,1] => [[5,5,2],[4,1]] => [4,1] => 2
010010 => [2,3,2] => [[5,4,2],[3,1]] => [3,1] => 2
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => 2
010100 => [2,2,3] => [[5,3,2],[2,1]] => [2,1] => 2
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => 3
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => 2
011000 => [2,1,4] => [[5,2,2],[1,1]] => [1,1] => 1
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => 2
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => 2
011011 => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => 2
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => 2
011110 => [2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
011111 => [2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => 1
100000 => [1,6] => [[6,1],[]] => [] => 0
100001 => [1,5,1] => [[5,5,1],[4]] => [4] => 1
100010 => [1,4,2] => [[5,4,1],[3]] => [3] => 1
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => 1
100100 => [1,3,3] => [[5,3,1],[2]] => [2] => 1
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => 2
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => 1
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Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the 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.
Map
inner shape
Description
The inner shape of a skew partition.
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