Identifier
-
Mp00093:
Dyck paths
—to binary word⟶
Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000764: Integer compositions ⟶ 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] => 1
[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] => [2,1,1,2] => 1
[1,1,0,0,1,0] => 110010 => [2,2,1,1] => [1,2,2,1] => 2
[1,1,0,1,0,0] => 110100 => [2,1,1,2] => [2,2,1,1] => 1
[1,1,1,0,0,0] => 111000 => [3,3] => [3,3] => 1
[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] => [2,1,1,1,1,2] => 1
[1,0,1,1,0,0,1,0] => 10110010 => [1,1,2,2,1,1] => [1,1,1,2,2,1] => 2
[1,0,1,1,0,1,0,0] => 10110100 => [1,1,2,1,1,2] => [2,1,1,2,1,1] => 1
[1,0,1,1,1,0,0,0] => 10111000 => [1,1,3,3] => [3,1,1,3] => 1
[1,1,0,0,1,0,1,0] => 11001010 => [2,2,1,1,1,1] => [1,2,2,1,1,1] => 2
[1,1,0,0,1,1,0,0] => 11001100 => [2,2,2,2] => [2,2,2,2] => 1
[1,1,0,1,0,0,1,0] => 11010010 => [2,1,1,2,1,1] => [1,2,1,1,2,1] => 2
[1,1,0,1,0,1,0,0] => 11010100 => [2,1,1,1,1,2] => [2,2,1,1,1,1] => 1
[1,1,0,1,1,0,0,0] => 11011000 => [2,1,2,3] => [3,2,1,2] => 1
[1,1,1,0,0,0,1,0] => 11100010 => [3,3,1,1] => [1,3,3,1] => 2
[1,1,1,0,0,1,0,0] => 11100100 => [3,2,1,2] => [2,3,2,1] => 2
[1,1,1,0,1,0,0,0] => 11101000 => [3,1,1,3] => [3,3,1,1] => 1
[1,1,1,1,0,0,0,0] => 11110000 => [4,4] => [4,4] => 1
[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] => [2,1,1,1,1,1,1,2] => 1
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => [1,1,1,1,2,1,1,2] => [2,1,1,1,1,2,1,1] => 1
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => [1,1,1,1,3,3] => [3,1,1,1,1,3] => 1
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => [1,1,2,2,2,2] => [2,1,1,2,2,2] => 1
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => [1,1,2,1,1,1,1,2] => [2,1,1,2,1,1,1,1] => 1
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => [1,1,2,1,2,3] => [3,1,1,2,1,2] => 1
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => [1,1,3,1,1,3] => [3,1,1,3,1,1] => 1
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => [1,1,4,4] => [4,1,1,4] => 1
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => [2,2,1,1,2,2] => [2,2,2,1,1,2] => 1
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => [2,2,2,1,1,2] => [2,2,2,2,1,1] => 1
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => [2,2,3,3] => [3,2,2,3] => 1
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => [2,1,1,2,2,2] => [2,2,1,1,2,2] => 1
[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] => 1
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => [2,1,1,1,2,3] => [3,2,1,1,1,2] => 1
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => [2,1,2,1,1,3] => [3,2,1,2,1,1] => 1
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => [2,1,3,4] => [4,2,1,3] => 1
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => [3,2,2,3] => [3,3,2,2] => 1
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => [3,1,1,3,1,1] => [1,3,1,1,3,1] => 2
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => [3,1,1,1,1,3] => [3,3,1,1,1,1] => 1
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => [3,1,2,4] => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => [4,4,1,1] => [1,4,4,1] => 2
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => [4,1,1,4] => [4,4,1,1] => 1
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => [5,5] => [5,5] => 1
[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] => [2,1,1,1,1,1,1,1,1,2] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => [1,1,1,1,1,1,2,1,1,2] => [2,1,1,1,1,1,1,2,1,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => [1,1,1,1,1,1,3,3] => [3,1,1,1,1,1,1,3] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => [1,1,1,1,2,2,2,2] => [2,1,1,1,1,2,2,2] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => [1,1,1,1,2,1,1,1,1,2] => [2,1,1,1,1,2,1,1,1,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => [1,1,1,1,2,1,2,3] => [3,1,1,1,1,2,1,2] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => [1,1,1,1,3,1,1,3] => [3,1,1,1,1,3,1,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => [1,1,1,1,4,4] => [4,1,1,1,1,4] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => [1,1,2,2,1,1,2,2] => [2,1,1,2,2,1,1,2] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => [1,1,2,2,2,1,1,2] => [2,1,1,2,2,2,1,1] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => [1,1,2,2,3,3] => [3,1,1,2,2,3] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => [1,1,2,1,1,2,2,2] => [2,1,1,2,1,1,2,2] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => [1,1,2,1,1,1,1,1,1,2] => [2,1,1,2,1,1,1,1,1,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => [1,1,2,1,1,1,2,3] => [3,1,1,2,1,1,1,2] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => [1,1,2,1,2,1,1,3] => [3,1,1,2,1,2,1,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => [1,1,2,1,3,4] => [4,1,1,2,1,3] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => [1,1,3,2,2,3] => [3,1,1,3,2,2] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => [1,1,3,1,1,1,1,3] => [3,1,1,3,1,1,1,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => [1,1,3,1,2,4] => [4,1,1,3,1,2] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => [1,1,4,1,1,4] => [4,1,1,4,1,1] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => [1,1,5,5] => [5,1,1,5] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => [2,2,1,1,1,1,2,2] => [2,2,2,1,1,1,1,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => [2,2,1,1,2,1,1,2] => [2,2,2,1,1,2,1,1] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => [2,2,1,1,3,3] => [3,2,2,1,1,3] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => [2,2,2,2,2,2] => [2,2,2,2,2,2] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => [2,2,2,1,1,1,1,2] => [2,2,2,2,1,1,1,1] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => [2,2,2,1,2,3] => [3,2,2,2,1,2] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => [2,2,3,1,1,3] => [3,2,2,3,1,1] => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => [2,2,4,4] => [4,2,2,4] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => [2,1,1,2,1,1,2,2] => [2,2,1,1,2,1,1,2] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => [2,1,1,2,2,1,1,2] => [2,2,1,1,2,2,1,1] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => [2,1,1,2,3,3] => [3,2,1,1,2,3] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => [2,1,1,1,1,2,2,2] => [2,2,1,1,1,1,2,2] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => [2,1,1,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => [2,1,1,1,1,1,2,3] => [3,2,1,1,1,1,1,2] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => [2,1,1,1,2,1,1,3] => [3,2,1,1,1,2,1,1] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => [2,1,1,1,3,4] => [4,2,1,1,1,3] => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => [2,1,2,2,2,3] => [3,2,1,2,2,2] => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => [2,1,2,1,1,1,1,3] => [3,2,1,2,1,1,1,1] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => [2,1,2,1,2,4] => [4,2,1,2,1,2] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => [2,1,3,1,1,4] => [4,2,1,3,1,1] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => [2,1,4,5] => [5,2,1,4] => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => [3,3,3,3] => [3,3,3,3] => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => [3,2,1,1,2,3] => [3,3,2,1,1,2] => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => [3,2,2,1,1,3] => [3,3,2,2,1,1] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => [3,2,3,4] => [4,3,2,3] => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => [3,1,1,2,2,3] => [3,3,1,1,2,2] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => [3,1,1,1,1,1,1,3] => [3,3,1,1,1,1,1,1] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => [3,1,1,1,2,4] => [4,3,1,1,1,2] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => [3,1,2,1,1,4] => [4,3,1,2,1,1] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => [3,1,3,5] => [5,3,1,3] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => [4,2,2,4] => [4,4,2,2] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => [4,1,1,1,1,4] => [4,4,1,1,1,1] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => [4,1,2,5] => [5,4,1,2] => 1
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Description
The number of strong records in an integer composition.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record.
Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
rotate back to front
Description
The back to front rotation of an integer composition.
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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