- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001382: Integer partitions ⟶ ℤ

Values

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

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

00 => [2] => [2] => 1

01 => [1,1] => [1,1] => 1

10 => [1,1] => [1,1] => 1

11 => [2] => [2] => 1

000 => [3] => [3] => 2

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

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

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

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

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

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

111 => [3] => [3] => 2

0000 => [4] => [4] => 3

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

0010 => [2,1,1] => [2,1,1] => 3

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

0100 => [1,1,2] => [2,1,1] => 3

0101 => [1,1,1,1] => [1,1,1,1] => 3

0110 => [1,2,1] => [2,1,1] => 3

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

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

1001 => [1,2,1] => [2,1,1] => 3

1010 => [1,1,1,1] => [1,1,1,1] => 3

1011 => [1,1,2] => [2,1,1] => 3

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

1101 => [2,1,1] => [2,1,1] => 3

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

1111 => [4] => [4] => 3

00000 => [5] => [5] => 4

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

00010 => [3,1,1] => [3,1,1] => 4

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

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

00101 => [2,1,1,1] => [2,1,1,1] => 4

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

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

01000 => [1,1,3] => [3,1,1] => 4

01001 => [1,1,2,1] => [2,1,1,1] => 4

01010 => [1,1,1,1,1] => [1,1,1,1,1] => 4

01011 => [1,1,1,2] => [2,1,1,1] => 4

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

01101 => [1,2,1,1] => [2,1,1,1] => 4

01110 => [1,3,1] => [3,1,1] => 4

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

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

10001 => [1,3,1] => [3,1,1] => 4

10010 => [1,2,1,1] => [2,1,1,1] => 4

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

10100 => [1,1,1,2] => [2,1,1,1] => 4

10101 => [1,1,1,1,1] => [1,1,1,1,1] => 4

10110 => [1,1,2,1] => [2,1,1,1] => 4

10111 => [1,1,3] => [3,1,1] => 4

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

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

11010 => [2,1,1,1] => [2,1,1,1] => 4

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

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

11101 => [3,1,1] => [3,1,1] => 4

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

11111 => [5] => [5] => 4

000000 => [6] => [6] => 5

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

000010 => [4,1,1] => [4,1,1] => 5

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

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

000101 => [3,1,1,1] => [3,1,1,1] => 5

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

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

001000 => [2,1,3] => [3,2,1] => 2

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

001010 => [2,1,1,1,1] => [2,1,1,1,1] => 5

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

001100 => [2,2,2] => [2,2,2] => 2

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

001110 => [2,3,1] => [3,2,1] => 2

001111 => [2,4] => [4,2] => 2

010000 => [1,1,4] => [4,1,1] => 5

010001 => [1,1,3,1] => [3,1,1,1] => 5

010010 => [1,1,2,1,1] => [2,1,1,1,1] => 5

010011 => [1,1,2,2] => [2,2,1,1] => 2

010100 => [1,1,1,1,2] => [2,1,1,1,1] => 5

010101 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 5

010110 => [1,1,1,2,1] => [2,1,1,1,1] => 5

010111 => [1,1,1,3] => [3,1,1,1] => 5

011000 => [1,2,3] => [3,2,1] => 2

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

011010 => [1,2,1,1,1] => [2,1,1,1,1] => 5

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

011100 => [1,3,2] => [3,2,1] => 2

011101 => [1,3,1,1] => [3,1,1,1] => 5

011110 => [1,4,1] => [4,1,1] => 5

011111 => [1,5] => [5,1] => 5

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

100001 => [1,4,1] => [4,1,1] => 5

100010 => [1,3,1,1] => [3,1,1,1] => 5

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

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

100101 => [1,2,1,1,1] => [2,1,1,1,1] => 5

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

>>> Load all 2416 entries. <<<

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$F_{1} = 2$

$F_{2} = 4\ q$

$F_{3} = 8\ q^{2}$

$F_{4} = 2 + 14\ q^{3}$

$F_{5} = 10\ q + 22\ q^{4}$

$F_{6} = 32\ q^{2} + 32\ q^{5}$

$F_{7} = 84\ q^{3} + 44\ q^{6}$

$F_{8} = 198\ q^{4} + 58\ q^{7}$

$F_{9} = 2 + 436\ q^{5} + 74\ q^{8}$

Description

The number of boxes in the diagram of a partition that do not lie in its Durfee square.

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

$w$ .