- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤ

Values

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

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

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

01 => [1,1] => [2] => [2] => 2

10 => [1,1] => [2] => [2] => 2

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

000 => [3] => [1,1,1] => [1,1,1] => 1

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

010 => [1,1,1] => [3] => [3] => 3

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

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

101 => [1,1,1] => [3] => [3] => 3

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

111 => [3] => [1,1,1] => [1,1,1] => 1

0000 => [4] => [1,1,1,1] => [1,1,1,1] => 1

0001 => [3,1] => [1,1,2] => [2,1,1] => 2

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

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

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

0101 => [1,1,1,1] => [4] => [4] => 4

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

0111 => [1,3] => [2,1,1] => [2,1,1] => 2

1000 => [1,3] => [2,1,1] => [2,1,1] => 2

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

1010 => [1,1,1,1] => [4] => [4] => 4

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

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

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

1110 => [3,1] => [1,1,2] => [2,1,1] => 2

1111 => [4] => [1,1,1,1] => [1,1,1,1] => 1

00000 => [5] => [1,1,1,1,1] => [1,1,1,1,1] => 1

00001 => [4,1] => [1,1,1,2] => [2,1,1,1] => 2

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

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

00100 => [2,1,2] => [1,3,1] => [3,1,1] => 3

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

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

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

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

01001 => [1,1,2,1] => [3,2] => [3,2] => 3

01010 => [1,1,1,1,1] => [5] => [5] => 5

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

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

01101 => [1,2,1,1] => [2,3] => [3,2] => 3

01110 => [1,3,1] => [2,1,2] => [2,2,1] => 2

01111 => [1,4] => [2,1,1,1] => [2,1,1,1] => 2

10000 => [1,4] => [2,1,1,1] => [2,1,1,1] => 2

10001 => [1,3,1] => [2,1,2] => [2,2,1] => 2

10010 => [1,2,1,1] => [2,3] => [3,2] => 3

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

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

10101 => [1,1,1,1,1] => [5] => [5] => 5

10110 => [1,1,2,1] => [3,2] => [3,2] => 3

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

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

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

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

11011 => [2,1,2] => [1,3,1] => [3,1,1] => 3

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

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

11110 => [4,1] => [1,1,1,2] => [2,1,1,1] => 2

11111 => [5] => [1,1,1,1,1] => [1,1,1,1,1] => 1

000000 => [6] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 1

000001 => [5,1] => [1,1,1,1,2] => [2,1,1,1,1] => 2

000010 => [4,1,1] => [1,1,1,3] => [3,1,1,1] => 3

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

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

000101 => [3,1,1,1] => [1,1,4] => [4,1,1] => 4

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

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

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

001001 => [2,1,2,1] => [1,3,2] => [3,2,1] => 3

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

001011 => [2,1,1,2] => [1,4,1] => [4,1,1] => 4

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

001101 => [2,2,1,1] => [1,2,3] => [3,2,1] => 3

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

001111 => [2,4] => [1,2,1,1,1] => [2,1,1,1,1] => 2

010000 => [1,1,4] => [3,1,1,1] => [3,1,1,1] => 3

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

010010 => [1,1,2,1,1] => [3,3] => [3,3] => 3

010011 => [1,1,2,2] => [3,2,1] => [3,2,1] => 3

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

010101 => [1,1,1,1,1,1] => [6] => [6] => 6

010110 => [1,1,1,2,1] => [4,2] => [4,2] => 4

010111 => [1,1,1,3] => [4,1,1] => [4,1,1] => 4

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

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

011010 => [1,2,1,1,1] => [2,4] => [4,2] => 4

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

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

011101 => [1,3,1,1] => [2,1,3] => [3,2,1] => 3

011110 => [1,4,1] => [2,1,1,2] => [2,2,1,1] => 2

011111 => [1,5] => [2,1,1,1,1] => [2,1,1,1,1] => 2

100000 => [1,5] => [2,1,1,1,1] => [2,1,1,1,1] => 2

100001 => [1,4,1] => [2,1,1,2] => [2,2,1,1] => 2

100010 => [1,3,1,1] => [2,1,3] => [3,2,1] => 3

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

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

100101 => [1,2,1,1,1] => [2,4] => [4,2] => 4

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

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$F_{1} = 2\ q$

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

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

$F_{4} = 2\ q + 8\ q^{2} + 4\ q^{3} + 2\ q^{4}$

$F_{5} = 2\ q + 14\ q^{2} + 10\ q^{3} + 4\ q^{4} + 2\ q^{5}$

$F_{6} = 2\ q + 24\ q^{2} + 22\ q^{3} + 10\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{7} = 2\ q + 40\ q^{2} + 46\ q^{3} + 24\ q^{4} + 10\ q^{5} + 4\ q^{6} + 2\ q^{7}$

$F_{8} = 2\ q + 66\ q^{2} + 94\ q^{3} + 54\ q^{4} + 24\ q^{5} + 10\ q^{6} + 4\ q^{7} + 2\ q^{8}$

$F_{9} = 2\ q + 108\ q^{2} + 188\ q^{3} + 118\ q^{4} + 56\ q^{5} + 24\ q^{6} + 10\ q^{7} + 4\ q^{8} + 2\ q^{9}$

Description

The largest part of an integer partition.

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

to partition

Description

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

Map

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .