Identifier
-
Mp00095:
Integer partitions
—to binary word⟶
Binary words
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤ
Values
[1] => 10 => 00 => [2] => 2
[2] => 100 => 000 => [3] => 3
[1,1] => 110 => 010 => [1,1,1] => 1
[3] => 1000 => 0000 => [4] => 4
[2,1] => 1010 => 0010 => [2,1,1] => 2
[1,1,1] => 1110 => 0110 => [1,2,1] => 1
[4] => 10000 => 00000 => [5] => 5
[3,1] => 10010 => 00010 => [3,1,1] => 3
[2,2] => 1100 => 0100 => [1,1,2] => 1
[2,1,1] => 10110 => 00110 => [2,2,1] => 2
[1,1,1,1] => 11110 => 01110 => [1,3,1] => 1
[5] => 100000 => 000000 => [6] => 6
[4,1] => 100010 => 000010 => [4,1,1] => 4
[3,2] => 10100 => 00100 => [2,1,2] => 2
[3,1,1] => 100110 => 000110 => [3,2,1] => 3
[2,2,1] => 11010 => 01010 => [1,1,1,1,1] => 1
[2,1,1,1] => 101110 => 001110 => [2,3,1] => 2
[1,1,1,1,1] => 111110 => 011110 => [1,4,1] => 1
[6] => 1000000 => 0000000 => [7] => 7
[5,1] => 1000010 => 0000010 => [5,1,1] => 5
[4,2] => 100100 => 000100 => [3,1,2] => 3
[4,1,1] => 1000110 => 0000110 => [4,2,1] => 4
[3,3] => 11000 => 01000 => [1,1,3] => 1
[3,2,1] => 101010 => 001010 => [2,1,1,1,1] => 2
[3,1,1,1] => 1001110 => 0001110 => [3,3,1] => 3
[2,2,2] => 11100 => 01100 => [1,2,2] => 1
[2,2,1,1] => 110110 => 010110 => [1,1,1,2,1] => 1
[2,1,1,1,1] => 1011110 => 0011110 => [2,4,1] => 2
[1,1,1,1,1,1] => 1111110 => 0111110 => [1,5,1] => 1
[7] => 10000000 => 00000000 => [8] => 8
[6,1] => 10000010 => 00000010 => [6,1,1] => 6
[5,2] => 1000100 => 0000100 => [4,1,2] => 4
[5,1,1] => 10000110 => 00000110 => [5,2,1] => 5
[4,3] => 101000 => 001000 => [2,1,3] => 2
[4,2,1] => 1001010 => 0001010 => [3,1,1,1,1] => 3
[4,1,1,1] => 10001110 => 00001110 => [4,3,1] => 4
[3,3,1] => 110010 => 010010 => [1,1,2,1,1] => 1
[3,2,2] => 101100 => 001100 => [2,2,2] => 2
[3,2,1,1] => 1010110 => 0010110 => [2,1,1,2,1] => 2
[3,1,1,1,1] => 10011110 => 00011110 => [3,4,1] => 3
[2,2,2,1] => 111010 => 011010 => [1,2,1,1,1] => 1
[2,2,1,1,1] => 1101110 => 0101110 => [1,1,1,3,1] => 1
[2,1,1,1,1,1] => 10111110 => 00111110 => [2,5,1] => 2
[1,1,1,1,1,1,1] => 11111110 => 01111110 => [1,6,1] => 1
[8] => 100000000 => 000000000 => [9] => 9
[7,1] => 100000010 => 000000010 => [7,1,1] => 7
[6,2] => 10000100 => 00000100 => [5,1,2] => 5
[6,1,1] => 100000110 => 000000110 => [6,2,1] => 6
[5,3] => 1001000 => 0001000 => [3,1,3] => 3
[5,2,1] => 10001010 => 00001010 => [4,1,1,1,1] => 4
[5,1,1,1] => 100001110 => 000001110 => [5,3,1] => 5
[4,4] => 110000 => 010000 => [1,1,4] => 1
[4,3,1] => 1010010 => 0010010 => [2,1,2,1,1] => 2
[4,2,2] => 1001100 => 0001100 => [3,2,2] => 3
[4,2,1,1] => 10010110 => 00010110 => [3,1,1,2,1] => 3
[4,1,1,1,1] => 100011110 => 000011110 => [4,4,1] => 4
[3,3,2] => 110100 => 010100 => [1,1,1,1,2] => 1
[3,3,1,1] => 1100110 => 0100110 => [1,1,2,2,1] => 1
[3,2,2,1] => 1011010 => 0011010 => [2,2,1,1,1] => 2
[3,2,1,1,1] => 10101110 => 00101110 => [2,1,1,3,1] => 2
[3,1,1,1,1,1] => 100111110 => 000111110 => [3,5,1] => 3
[2,2,2,2] => 111100 => 011100 => [1,3,2] => 1
[2,2,2,1,1] => 1110110 => 0110110 => [1,2,1,2,1] => 1
[2,2,1,1,1,1] => 11011110 => 01011110 => [1,1,1,4,1] => 1
[2,1,1,1,1,1,1] => 101111110 => 001111110 => [2,6,1] => 2
[1,1,1,1,1,1,1,1] => 111111110 => 011111110 => [1,7,1] => 1
[9] => 1000000000 => 0000000000 => [10] => 10
[8,1] => 1000000010 => 0000000010 => [8,1,1] => 8
[7,2] => 100000100 => 000000100 => [6,1,2] => 6
[7,1,1] => 1000000110 => 0000000110 => [7,2,1] => 7
[6,3] => 10001000 => 00001000 => [4,1,3] => 4
[6,2,1] => 100001010 => 000001010 => [5,1,1,1,1] => 5
[6,1,1,1] => 1000001110 => 0000001110 => [6,3,1] => 6
[5,4] => 1010000 => 0010000 => [2,1,4] => 2
[5,3,1] => 10010010 => 00010010 => [3,1,2,1,1] => 3
[5,2,2] => 10001100 => 00001100 => [4,2,2] => 4
[5,2,1,1] => 100010110 => 000010110 => [4,1,1,2,1] => 4
[5,1,1,1,1] => 1000011110 => 0000011110 => [5,4,1] => 5
[4,4,1] => 1100010 => 0100010 => [1,1,3,1,1] => 1
[4,3,2] => 1010100 => 0010100 => [2,1,1,1,2] => 2
[4,3,1,1] => 10100110 => 00100110 => [2,1,2,2,1] => 2
[4,2,2,1] => 10011010 => 00011010 => [3,2,1,1,1] => 3
[4,2,1,1,1] => 100101110 => 000101110 => [3,1,1,3,1] => 3
[4,1,1,1,1,1] => 1000111110 => 0000111110 => [4,5,1] => 4
[3,3,3] => 111000 => 011000 => [1,2,3] => 1
[3,3,2,1] => 1101010 => 0101010 => [1,1,1,1,1,1,1] => 1
[3,3,1,1,1] => 11001110 => 01001110 => [1,1,2,3,1] => 1
[3,2,2,2] => 1011100 => 0011100 => [2,3,2] => 2
[3,2,2,1,1] => 10110110 => 00110110 => [2,2,1,2,1] => 2
[3,2,1,1,1,1] => 101011110 => 001011110 => [2,1,1,4,1] => 2
[2,2,2,2,1] => 1111010 => 0111010 => [1,3,1,1,1] => 1
[2,2,2,1,1,1] => 11101110 => 01101110 => [1,2,1,3,1] => 1
[2,2,1,1,1,1,1] => 110111110 => 010111110 => [1,1,1,5,1] => 1
[1,1,1,1,1,1,1,1,1] => 1111111110 => 0111111110 => [1,8,1] => 1
[7,3] => 100001000 => 000001000 => [5,1,3] => 5
[7,2,1] => 1000001010 => 0000001010 => [6,1,1,1,1] => 6
[6,4] => 10010000 => 00010000 => [3,1,4] => 3
[6,3,1] => 100010010 => 000010010 => [4,1,2,1,1] => 4
[6,2,2] => 100001100 => 000001100 => [5,2,2] => 5
[5,5] => 1100000 => 0100000 => [1,1,5] => 1
[5,4,1] => 10100010 => 00100010 => [2,1,3,1,1] => 2
>>> Load all 284 entries. <<<
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Description
The first part of an integer composition.
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
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
Map
twist
Description
Return the binary word with first letter inverted.
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