Identifier
-
Mp00093:
Dyck paths
—to binary word⟶
Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤ
Values
[1,0] => 10 => [1,1] => [1,1] => 2
[1,0,1,0] => 1010 => [1,1,1,1] => [1,1,1,1] => 4
[1,1,0,0] => 1100 => [2,2] => [2,2] => 0
[1,0,1,0,1,0] => 101010 => [1,1,1,1,1,1] => [1,1,1,1,1,1] => 6
[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] => 0
[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] => 8
[1,0,1,0,1,1,0,0] => 10101100 => [1,1,1,1,2,2] => [2,2,1,1,1,1] => 4
[1,0,1,1,0,0,1,0] => 10110010 => [1,1,2,2,1,1] => [2,2,1,1,1,1] => 4
[1,0,1,1,0,1,0,0] => 10110100 => [1,1,2,1,1,2] => [2,2,1,1,1,1] => 4
[1,0,1,1,1,0,0,0] => 10111000 => [1,1,3,3] => [3,3,1,1] => 2
[1,1,0,0,1,0,1,0] => 11001010 => [2,2,1,1,1,1] => [2,2,1,1,1,1] => 4
[1,1,0,0,1,1,0,0] => 11001100 => [2,2,2,2] => [2,2,2,2] => 0
[1,1,0,1,0,0,1,0] => 11010010 => [2,1,1,2,1,1] => [2,2,1,1,1,1] => 4
[1,1,0,1,0,1,0,0] => 11010100 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => 4
[1,1,0,1,1,0,0,0] => 11011000 => [2,1,2,3] => [3,2,2,1] => 1
[1,1,1,0,0,0,1,0] => 11100010 => [3,3,1,1] => [3,3,1,1] => 2
[1,1,1,0,0,1,0,0] => 11100100 => [3,2,1,2] => [3,2,2,1] => 1
[1,1,1,0,1,0,0,0] => 11101000 => [3,1,1,3] => [3,3,1,1] => 2
[1,1,1,1,0,0,0,0] => 11110000 => [4,4] => [4,4] => 0
[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] => 10
[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] => 6
[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] => 6
[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] => 6
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => [1,1,1,1,3,3] => [3,3,1,1,1,1] => 4
[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] => 6
[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] => 6
[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] => 6
[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] => 4
[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] => 4
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => [1,1,4,4] => [4,4,1,1] => 2
[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] => 6
[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] => 0
[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] => 6
[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] => 6
[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] => 6
[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] => 1
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => [3,3,1,1,1,1] => [3,3,1,1,1,1] => 4
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => [3,3,2,2] => [3,3,2,2] => 0
[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] => 0
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => [3,1,1,3,1,1] => [3,3,1,1,1,1] => 4
[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] => 4
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => [3,1,2,4] => [4,3,2,1] => 1
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => [4,4,1,1] => [4,4,1,1] => 2
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => [4,3,1,2] => [4,3,2,1] => 1
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => [4,2,1,3] => [4,3,2,1] => 1
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => [4,1,1,4] => [4,4,1,1] => 2
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => [5,5] => [5,5] => 0
[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] => 12
[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] => 8
[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] => 8
[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] => 8
[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] => 6
[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] => 8
[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] => 4
[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] => 8
[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] => 8
[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] => 5
[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] => 6
[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] => 5
[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] => 6
[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] => 8
[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] => 4
[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] => 4
[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] => 4
[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] => 2
[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] => 8
[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] => 4
[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] => 8
[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] => 8
[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] => 5
[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] => 5
[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] => 4
[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] => 5
[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] => 3
[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] => 6
[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] => 2
[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] => 5
[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] => 5
[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] => 2
[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] => 6
[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] => 5
[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] => 6
[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] => 3
>>> Load all 196 entries. <<<
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Description
The number of parts equal to 1 in a partition.
Map
to binary word
Description
Return the Dyck word as binary word.
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 $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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