- FindStat

Identifier

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤ

Values

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

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

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

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

[1,0,1,1,0,0] => 101100 => [1,1,2,2] => [2,2,1,1] => 2

[1,1,0,0,1,0] => 110010 => [2,2,1,1] => [2,2,1,1] => 2

[1,1,0,1,0,0] => 110100 => [2,1,1,2] => [2,2,1,1] => 2

[1,1,1,0,0,0] => 111000 => [3,3] => [3,3] => 3

[1,0,1,0,1,0,1,0] => 10101010 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 1

[1,0,1,0,1,1,0,0] => 10101100 => [1,1,1,1,2,2] => [2,2,1,1,1,1] => 2

[1,0,1,1,0,0,1,0] => 10110010 => [1,1,2,2,1,1] => [2,2,1,1,1,1] => 2

[1,0,1,1,0,1,0,0] => 10110100 => [1,1,2,1,1,2] => [2,2,1,1,1,1] => 2

[1,0,1,1,1,0,0,0] => 10111000 => [1,1,3,3] => [3,3,1,1] => 3

[1,1,0,0,1,0,1,0] => 11001010 => [2,2,1,1,1,1] => [2,2,1,1,1,1] => 2

[1,1,0,0,1,1,0,0] => 11001100 => [2,2,2,2] => [2,2,2,2] => 2

[1,1,0,1,0,0,1,0] => 11010010 => [2,1,1,2,1,1] => [2,2,1,1,1,1] => 2

[1,1,0,1,0,1,0,0] => 11010100 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => 2

[1,1,0,1,1,0,0,0] => 11011000 => [2,1,2,3] => [3,2,2,1] => 3

[1,1,1,0,0,0,1,0] => 11100010 => [3,3,1,1] => [3,3,1,1] => 3

[1,1,1,0,0,1,0,0] => 11100100 => [3,2,1,2] => [3,2,2,1] => 3

[1,1,1,0,1,0,0,0] => 11101000 => [3,1,1,3] => [3,3,1,1] => 3

[1,1,1,1,0,0,0,0] => 11110000 => [4,4] => [4,4] => 4

[1,0,1,0,1,0,1,0,1,0] => 1010101010 => [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 1

[1,0,1,0,1,0,1,1,0,0] => 1010101100 => [1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1] => 2

[1,0,1,0,1,1,0,0,1,0] => 1010110010 => [1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1] => 2

[1,0,1,0,1,1,0,1,0,0] => 1010110100 => [1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1] => 2

[1,0,1,0,1,1,1,0,0,0] => 1010111000 => [1,1,1,1,3,3] => [3,3,1,1,1,1] => 3

[1,0,1,1,0,0,1,0,1,0] => 1011001010 => [1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 2

[1,0,1,1,0,0,1,1,0,0] => 1011001100 => [1,1,2,2,2,2] => [2,2,2,2,1,1] => 2

[1,0,1,1,0,1,0,0,1,0] => 1011010010 => [1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 2

[1,0,1,1,0,1,0,1,0,0] => 1011010100 => [1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 2

[1,0,1,1,0,1,1,0,0,0] => 1011011000 => [1,1,2,1,2,3] => [3,2,2,1,1,1] => 3

[1,0,1,1,1,0,0,0,1,0] => 1011100010 => [1,1,3,3,1,1] => [3,3,1,1,1,1] => 3

[1,0,1,1,1,0,0,1,0,0] => 1011100100 => [1,1,3,2,1,2] => [3,2,2,1,1,1] => 3

[1,0,1,1,1,0,1,0,0,0] => 1011101000 => [1,1,3,1,1,3] => [3,3,1,1,1,1] => 3

[1,0,1,1,1,1,0,0,0,0] => 1011110000 => [1,1,4,4] => [4,4,1,1] => 4

[1,1,0,0,1,0,1,0,1,0] => 1100101010 => [2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 2

[1,1,0,0,1,0,1,1,0,0] => 1100101100 => [2,2,1,1,2,2] => [2,2,2,2,1,1] => 2

[1,1,0,0,1,1,0,0,1,0] => 1100110010 => [2,2,2,2,1,1] => [2,2,2,2,1,1] => 2

[1,1,0,0,1,1,0,1,0,0] => 1100110100 => [2,2,2,1,1,2] => [2,2,2,2,1,1] => 2

[1,1,0,0,1,1,1,0,0,0] => 1100111000 => [2,2,3,3] => [3,3,2,2] => 3

[1,1,0,1,0,0,1,0,1,0] => 1101001010 => [2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1] => 2

[1,1,0,1,0,0,1,1,0,0] => 1101001100 => [2,1,1,2,2,2] => [2,2,2,2,1,1] => 2

[1,1,0,1,0,1,0,0,1,0] => 1101010010 => [2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1] => 2

[1,1,0,1,0,1,0,1,0,0] => 1101010100 => [2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1] => 2

[1,1,0,1,0,1,1,0,0,0] => 1101011000 => [2,1,1,1,2,3] => [3,2,2,1,1,1] => 3

[1,1,0,1,1,0,0,0,1,0] => 1101100010 => [2,1,2,3,1,1] => [3,2,2,1,1,1] => 3

[1,1,0,1,1,0,0,1,0,0] => 1101100100 => [2,1,2,2,1,2] => [2,2,2,2,1,1] => 2

[1,1,0,1,1,0,1,0,0,0] => 1101101000 => [2,1,2,1,1,3] => [3,2,2,1,1,1] => 3

[1,1,0,1,1,1,0,0,0,0] => 1101110000 => [2,1,3,4] => [4,3,2,1] => 4

[1,1,1,0,0,0,1,0,1,0] => 1110001010 => [3,3,1,1,1,1] => [3,3,1,1,1,1] => 3

[1,1,1,0,0,0,1,1,0,0] => 1110001100 => [3,3,2,2] => [3,3,2,2] => 3

[1,1,1,0,0,1,0,0,1,0] => 1110010010 => [3,2,1,2,1,1] => [3,2,2,1,1,1] => 3

[1,1,1,0,0,1,0,1,0,0] => 1110010100 => [3,2,1,1,1,2] => [3,2,2,1,1,1] => 3

[1,1,1,0,0,1,1,0,0,0] => 1110011000 => [3,2,2,3] => [3,3,2,2] => 3

[1,1,1,0,1,0,0,0,1,0] => 1110100010 => [3,1,1,3,1,1] => [3,3,1,1,1,1] => 3

[1,1,1,0,1,0,0,1,0,0] => 1110100100 => [3,1,1,2,1,2] => [3,2,2,1,1,1] => 3

[1,1,1,0,1,0,1,0,0,0] => 1110101000 => [3,1,1,1,1,3] => [3,3,1,1,1,1] => 3

[1,1,1,0,1,1,0,0,0,0] => 1110110000 => [3,1,2,4] => [4,3,2,1] => 4

[1,1,1,1,0,0,0,0,1,0] => 1111000010 => [4,4,1,1] => [4,4,1,1] => 4

[1,1,1,1,0,0,0,1,0,0] => 1111000100 => [4,3,1,2] => [4,3,2,1] => 4

[1,1,1,1,0,0,1,0,0,0] => 1111001000 => [4,2,1,3] => [4,3,2,1] => 4

[1,1,1,1,0,1,0,0,0,0] => 1111010000 => [4,1,1,4] => [4,4,1,1] => 4

[1,1,1,1,1,0,0,0,0,0] => 1111100000 => [5,5] => [5,5] => 5

[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => [1,1,1,1,1,1,1,1,2,2] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => [1,1,1,1,1,1,2,2,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => [1,1,1,1,1,1,2,1,1,2] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => [1,1,1,1,1,1,3,3] => [3,3,1,1,1,1,1,1] => 3

[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => [1,1,1,1,2,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => [1,1,1,1,2,2,2,2] => [2,2,2,2,1,1,1,1] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => [1,1,1,1,2,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => [1,1,1,1,2,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => [1,1,1,1,2,1,2,3] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => [1,1,1,1,3,3,1,1] => [3,3,1,1,1,1,1,1] => 3

[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => [1,1,1,1,3,2,1,2] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => [1,1,1,1,3,1,1,3] => [3,3,1,1,1,1,1,1] => 3

[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => [1,1,1,1,4,4] => [4,4,1,1,1,1] => 4

[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => [1,1,2,2,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => [1,1,2,2,1,1,2,2] => [2,2,2,2,1,1,1,1] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => [1,1,2,2,2,2,1,1] => [2,2,2,2,1,1,1,1] => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => [1,1,2,2,2,1,1,2] => [2,2,2,2,1,1,1,1] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => [1,1,2,2,3,3] => [3,3,2,2,1,1] => 3

[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => [1,1,2,1,1,2,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => [1,1,2,1,1,2,2,2] => [2,2,2,2,1,1,1,1] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => [1,1,2,1,1,1,1,2,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => [1,1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1] => 2

[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => [1,1,2,1,1,1,2,3] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => [1,1,2,1,2,3,1,1] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => [1,1,2,1,2,2,1,2] => [2,2,2,2,1,1,1,1] => 2

[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => [1,1,2,1,2,1,1,3] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => [1,1,2,1,3,4] => [4,3,2,1,1,1] => 4

[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => [1,1,3,3,1,1,1,1] => [3,3,1,1,1,1,1,1] => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => [1,1,3,3,2,2] => [3,3,2,2,1,1] => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => [1,1,3,2,1,2,1,1] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => [1,1,3,2,1,1,1,2] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => [1,1,3,2,2,3] => [3,3,2,2,1,1] => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => [1,1,3,1,1,3,1,1] => [3,3,1,1,1,1,1,1] => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => [1,1,3,1,1,2,1,2] => [3,2,2,1,1,1,1,1] => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => [1,1,3,1,1,1,1,3] => [3,3,1,1,1,1,1,1] => 3

[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => [1,1,3,1,2,4] => [4,3,2,1,1,1] => 4

>>> Load all 196 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = q + 7\ q^{2} + 5\ q^{3} + q^{4}$

$F_{5} = q + 16\ q^{2} + 17\ q^{3} + 7\ q^{4} + q^{5}$

$F_{6} = q + 36\ q^{2} + 56\ q^{3} + 29\ q^{4} + 9\ q^{5} + q^{6}$

Description

The largest part of an integer partition.

Map

to partition

Description

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

Map

to binary word

Description

Return the Dyck word as binary word.

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

$w$ .