Identifier
-
Mp00095:
Integer partitions
—to binary word⟶
Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)
Values
[1] => 10 => [1,2] => [2,1] => 3
[2] => 100 => [1,3] => [3,1] => 3
[1,1] => 110 => [1,1,2] => [2,1,1] => 3
[3] => 1000 => [1,4] => [4,1] => 3
[2,1] => 1010 => [1,2,2] => [2,2,1] => 3
[1,1,1] => 1110 => [1,1,1,2] => [2,1,1,1] => 3
[4] => 10000 => [1,5] => [5,1] => 3
[3,1] => 10010 => [1,3,2] => [3,2,1] => 4
[2,2] => 1100 => [1,1,3] => [3,1,1] => 3
[2,1,1] => 10110 => [1,2,1,2] => [2,2,1,1] => 3
[1,1,1,1] => 11110 => [1,1,1,1,2] => [2,1,1,1,1] => 3
[5] => 100000 => [1,6] => [6,1] => 3
[4,1] => 100010 => [1,4,2] => [4,2,1] => 4
[3,2] => 10100 => [1,2,3] => [3,2,1] => 4
[3,1,1] => 100110 => [1,3,1,2] => [3,2,1,1] => 4
[2,2,1] => 11010 => [1,1,2,2] => [2,2,1,1] => 3
[2,1,1,1] => 101110 => [1,2,1,1,2] => [2,2,1,1,1] => 3
[1,1,1,1,1] => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1] => 3
[6] => 1000000 => [1,7] => [7,1] => 3
[5,1] => 1000010 => [1,5,2] => [5,2,1] => 4
[4,2] => 100100 => [1,3,3] => [3,3,1] => 3
[4,1,1] => 1000110 => [1,4,1,2] => [4,2,1,1] => 4
[3,3] => 11000 => [1,1,4] => [4,1,1] => 3
[3,2,1] => 101010 => [1,2,2,2] => [2,2,2,1] => 3
[3,1,1,1] => 1001110 => [1,3,1,1,2] => [3,2,1,1,1] => 4
[2,2,2] => 11100 => [1,1,1,3] => [3,1,1,1] => 3
[2,2,1,1] => 110110 => [1,1,2,1,2] => [2,2,1,1,1] => 3
[2,1,1,1,1] => 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1] => 3
[1,1,1,1,1,1] => 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => 3
[7] => 10000000 => [1,8] => [8,1] => 3
[6,1] => 10000010 => [1,6,2] => [6,2,1] => 4
[5,2] => 1000100 => [1,4,3] => [4,3,1] => 4
[5,1,1] => 10000110 => [1,5,1,2] => [5,2,1,1] => 4
[4,3] => 101000 => [1,2,4] => [4,2,1] => 4
[4,2,1] => 1001010 => [1,3,2,2] => [3,2,2,1] => 4
[4,1,1,1] => 10001110 => [1,4,1,1,2] => [4,2,1,1,1] => 4
[3,3,1] => 110010 => [1,1,3,2] => [3,2,1,1] => 4
[3,2,2] => 101100 => [1,2,1,3] => [3,2,1,1] => 4
[3,2,1,1] => 1010110 => [1,2,2,1,2] => [2,2,2,1,1] => 3
[3,1,1,1,1] => 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1] => 4
[2,2,2,1] => 111010 => [1,1,1,2,2] => [2,2,1,1,1] => 3
[2,2,1,1,1] => 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1] => 3
[2,1,1,1,1,1] => 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1] => 3
[1,1,1,1,1,1,1] => 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => 3
[8] => 100000000 => [1,9] => [9,1] => 3
[7,1] => 100000010 => [1,7,2] => [7,2,1] => 4
[6,2] => 10000100 => [1,5,3] => [5,3,1] => 4
[6,1,1] => 100000110 => [1,6,1,2] => [6,2,1,1] => 4
[5,3] => 1001000 => [1,3,4] => [4,3,1] => 4
[5,2,1] => 10001010 => [1,4,2,2] => [4,2,2,1] => 4
[5,1,1,1] => 100001110 => [1,5,1,1,2] => [5,2,1,1,1] => 4
[4,4] => 110000 => [1,1,5] => [5,1,1] => 3
[4,3,1] => 1010010 => [1,2,3,2] => [3,2,2,1] => 4
[4,2,2] => 1001100 => [1,3,1,3] => [3,3,1,1] => 3
[4,2,1,1] => 10010110 => [1,3,2,1,2] => [3,2,2,1,1] => 4
[4,1,1,1,1] => 100011110 => [1,4,1,1,1,2] => [4,2,1,1,1,1] => 4
[3,3,2] => 110100 => [1,1,2,3] => [3,2,1,1] => 4
[3,3,1,1] => 1100110 => [1,1,3,1,2] => [3,2,1,1,1] => 4
[3,2,2,1] => 1011010 => [1,2,1,2,2] => [2,2,2,1,1] => 3
[3,2,1,1,1] => 10101110 => [1,2,2,1,1,2] => [2,2,2,1,1,1] => 3
[3,1,1,1,1,1] => 100111110 => [1,3,1,1,1,1,2] => [3,2,1,1,1,1,1] => 4
[2,2,2,2] => 111100 => [1,1,1,1,3] => [3,1,1,1,1] => 3
[2,2,2,1,1] => 1110110 => [1,1,1,2,1,2] => [2,2,1,1,1,1] => 3
[2,2,1,1,1,1] => 11011110 => [1,1,2,1,1,1,2] => [2,2,1,1,1,1,1] => 3
[2,1,1,1,1,1,1] => 101111110 => [1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,1,1] => 111111110 => [1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => 3
[7,2] => 100000100 => [1,6,3] => [6,3,1] => 4
[6,3] => 10001000 => [1,4,4] => [4,4,1] => 3
[6,2,1] => 100001010 => [1,5,2,2] => [5,2,2,1] => 4
[5,4] => 1010000 => [1,2,5] => [5,2,1] => 4
[5,3,1] => 10010010 => [1,3,3,2] => [3,3,2,1] => 4
[5,2,2] => 10001100 => [1,4,1,3] => [4,3,1,1] => 4
[5,2,1,1] => 100010110 => [1,4,2,1,2] => [4,2,2,1,1] => 4
[4,4,1] => 1100010 => [1,1,4,2] => [4,2,1,1] => 4
[4,3,2] => 1010100 => [1,2,2,3] => [3,2,2,1] => 4
[4,3,1,1] => 10100110 => [1,2,3,1,2] => [3,2,2,1,1] => 4
[4,2,2,1] => 10011010 => [1,3,1,2,2] => [3,2,2,1,1] => 4
[4,2,1,1,1] => 100101110 => [1,3,2,1,1,2] => [3,2,2,1,1,1] => 4
[3,3,3] => 111000 => [1,1,1,4] => [4,1,1,1] => 3
[3,3,2,1] => 1101010 => [1,1,2,2,2] => [2,2,2,1,1] => 3
[3,3,1,1,1] => 11001110 => [1,1,3,1,1,2] => [3,2,1,1,1,1] => 4
[3,2,2,2] => 1011100 => [1,2,1,1,3] => [3,2,1,1,1] => 4
[3,2,2,1,1] => 10110110 => [1,2,1,2,1,2] => [2,2,2,1,1,1] => 3
[3,2,1,1,1,1] => 101011110 => [1,2,2,1,1,1,2] => [2,2,2,1,1,1,1] => 3
[2,2,2,2,1] => 1111010 => [1,1,1,1,2,2] => [2,2,1,1,1,1] => 3
[2,2,2,1,1,1] => 11101110 => [1,1,1,2,1,1,2] => [2,2,1,1,1,1,1] => 3
[2,2,1,1,1,1,1] => 110111110 => [1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 3
[7,3] => 100001000 => [1,5,4] => [5,4,1] => 4
[6,4] => 10010000 => [1,3,5] => [5,3,1] => 4
[6,3,1] => 100010010 => [1,4,3,2] => [4,3,2,1] => 5
[6,2,2] => 100001100 => [1,5,1,3] => [5,3,1,1] => 4
[6,2,1,1] => 1000010110 => [1,5,2,1,2] => [5,2,2,1,1] => 4
[5,5] => 1100000 => [1,1,6] => [6,1,1] => 3
[5,4,1] => 10100010 => [1,2,4,2] => [4,2,2,1] => 4
[5,3,2] => 10010100 => [1,3,2,3] => [3,3,2,1] => 4
[5,3,1,1] => 100100110 => [1,3,3,1,2] => [3,3,2,1,1] => 4
[5,2,2,1] => 100011010 => [1,4,1,2,2] => [4,2,2,1,1] => 4
[4,4,2] => 1100100 => [1,1,3,3] => [3,3,1,1] => 3
[4,4,1,1] => 11000110 => [1,1,4,1,2] => [4,2,1,1,1] => 4
[4,3,3] => 1011000 => [1,2,1,4] => [4,2,1,1] => 4
[4,3,2,1] => 10101010 => [1,2,2,2,2] => [2,2,2,2,1] => 3
>>> Load all 269 entries. <<<
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Description
The number of addable cells of the Ferrers diagram of an integer partition.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
to partition
Description
Sends a composition to the partition obtained by sorting the entries.
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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