- FindStat

Identifier

Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001525: Integer partitions ⟶ ℤ

Values

0 => 1 => [1] => [1] => 1

1 => 0 => [1] => [1] => 1

00 => 10 => [1,1] => [1,1] => 0

01 => 11 => [2] => [2] => 0

10 => 00 => [2] => [2] => 0

11 => 01 => [1,1] => [1,1] => 0

000 => 100 => [1,2] => [2,1] => 1

001 => 101 => [1,1,1] => [1,1,1] => 0

010 => 110 => [2,1] => [2,1] => 1

011 => 111 => [3] => [3] => 0

100 => 000 => [3] => [3] => 0

101 => 001 => [2,1] => [2,1] => 1

110 => 010 => [1,1,1] => [1,1,1] => 0

111 => 011 => [1,2] => [2,1] => 1

0000 => 1000 => [1,3] => [3,1] => 0

0001 => 1001 => [1,2,1] => [2,1,1] => 0

0010 => 1010 => [1,1,1,1] => [1,1,1,1] => 0

0011 => 1011 => [1,1,2] => [2,1,1] => 0

0100 => 1100 => [2,2] => [2,2] => 2

0101 => 1101 => [2,1,1] => [2,1,1] => 0

0110 => 1110 => [3,1] => [3,1] => 0

0111 => 1111 => [4] => [4] => 0

1000 => 0000 => [4] => [4] => 0

1001 => 0001 => [3,1] => [3,1] => 0

1010 => 0010 => [2,1,1] => [2,1,1] => 0

1011 => 0011 => [2,2] => [2,2] => 2

1100 => 0100 => [1,1,2] => [2,1,1] => 0

1101 => 0101 => [1,1,1,1] => [1,1,1,1] => 0

1110 => 0110 => [1,2,1] => [2,1,1] => 0

1111 => 0111 => [1,3] => [3,1] => 0

00000 => 10000 => [1,4] => [4,1] => 0

00001 => 10001 => [1,3,1] => [3,1,1] => 1

00010 => 10010 => [1,2,1,1] => [2,1,1,1] => 0

00011 => 10011 => [1,2,2] => [2,2,1] => 1

00100 => 10100 => [1,1,1,2] => [2,1,1,1] => 0

00101 => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 0

00110 => 10110 => [1,1,2,1] => [2,1,1,1] => 0

00111 => 10111 => [1,1,3] => [3,1,1] => 1

01000 => 11000 => [2,3] => [3,2] => 1

01001 => 11001 => [2,2,1] => [2,2,1] => 1

01010 => 11010 => [2,1,1,1] => [2,1,1,1] => 0

01011 => 11011 => [2,1,2] => [2,2,1] => 1

01100 => 11100 => [3,2] => [3,2] => 1

01101 => 11101 => [3,1,1] => [3,1,1] => 1

01110 => 11110 => [4,1] => [4,1] => 0

01111 => 11111 => [5] => [5] => 0

10000 => 00000 => [5] => [5] => 0

10001 => 00001 => [4,1] => [4,1] => 0

10010 => 00010 => [3,1,1] => [3,1,1] => 1

10011 => 00011 => [3,2] => [3,2] => 1

10100 => 00100 => [2,1,2] => [2,2,1] => 1

10101 => 00101 => [2,1,1,1] => [2,1,1,1] => 0

10110 => 00110 => [2,2,1] => [2,2,1] => 1

10111 => 00111 => [2,3] => [3,2] => 1

11000 => 01000 => [1,1,3] => [3,1,1] => 1

11001 => 01001 => [1,1,2,1] => [2,1,1,1] => 0

11010 => 01010 => [1,1,1,1,1] => [1,1,1,1,1] => 0

11011 => 01011 => [1,1,1,2] => [2,1,1,1] => 0

11100 => 01100 => [1,2,2] => [2,2,1] => 1

11101 => 01101 => [1,2,1,1] => [2,1,1,1] => 0

11110 => 01110 => [1,3,1] => [3,1,1] => 1

11111 => 01111 => [1,4] => [4,1] => 0

000000 => 100000 => [1,5] => [5,1] => 0

000001 => 100001 => [1,4,1] => [4,1,1] => 0

000010 => 100010 => [1,3,1,1] => [3,1,1,1] => 0

000011 => 100011 => [1,3,2] => [3,2,1] => 2

000100 => 100100 => [1,2,1,2] => [2,2,1,1] => 1

000101 => 100101 => [1,2,1,1,1] => [2,1,1,1,1] => 0

000110 => 100110 => [1,2,2,1] => [2,2,1,1] => 1

000111 => 100111 => [1,2,3] => [3,2,1] => 2

001000 => 101000 => [1,1,1,3] => [3,1,1,1] => 0

001001 => 101001 => [1,1,1,2,1] => [2,1,1,1,1] => 0

001010 => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 0

001011 => 101011 => [1,1,1,1,2] => [2,1,1,1,1] => 0

001100 => 101100 => [1,1,2,2] => [2,2,1,1] => 1

001101 => 101101 => [1,1,2,1,1] => [2,1,1,1,1] => 0

001110 => 101110 => [1,1,3,1] => [3,1,1,1] => 0

001111 => 101111 => [1,1,4] => [4,1,1] => 0

010000 => 110000 => [2,4] => [4,2] => 1

010001 => 110001 => [2,3,1] => [3,2,1] => 2

010010 => 110010 => [2,2,1,1] => [2,2,1,1] => 1

010011 => 110011 => [2,2,2] => [2,2,2] => 0

010100 => 110100 => [2,1,1,2] => [2,2,1,1] => 1

010101 => 110101 => [2,1,1,1,1] => [2,1,1,1,1] => 0

010110 => 110110 => [2,1,2,1] => [2,2,1,1] => 1

010111 => 110111 => [2,1,3] => [3,2,1] => 2

011000 => 111000 => [3,3] => [3,3] => 0

011001 => 111001 => [3,2,1] => [3,2,1] => 2

011010 => 111010 => [3,1,1,1] => [3,1,1,1] => 0

011011 => 111011 => [3,1,2] => [3,2,1] => 2

011100 => 111100 => [4,2] => [4,2] => 1

011101 => 111101 => [4,1,1] => [4,1,1] => 0

011110 => 111110 => [5,1] => [5,1] => 0

011111 => 111111 => [6] => [6] => 0

100000 => 000000 => [6] => [6] => 0

100001 => 000001 => [5,1] => [5,1] => 0

100010 => 000010 => [4,1,1] => [4,1,1] => 0

100011 => 000011 => [4,2] => [4,2] => 1

100100 => 000100 => [3,1,2] => [3,2,1] => 2

100101 => 000101 => [3,1,1,1] => [3,1,1,1] => 0

100110 => 000110 => [3,2,1] => [3,2,1] => 2

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$F_{1} = 2\ q$

$F_{2} = 4$

$F_{3} = 4 + 4\ q$

$F_{4} = 14 + 2\ q^{2}$

$F_{5} = 16 + 16\ q$

$F_{6} = 36 + 16\ q + 12\ q^{2}$

$F_{7} = 48 + 80\ q$

$F_{8} = 140 + 86\ q + 30\ q^{2}$

$F_{9} = 234 + 276\ q + 2\ q^{3}$

Description

The number of symmetric hooks on the diagonal of a partition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

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

$w$ .

Map

twist

Description

Return the binary word with first letter inverted.