Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000807: Integer compositions ⟶ ℤ
Values
[1] => [1,0] => 10 => [1,1] => 0
[1,1] => [1,0,1,0] => 1010 => [1,1,1,1] => 0
[2] => [1,1,0,0] => 1100 => [2,2] => 0
[1,1,1] => [1,0,1,0,1,0] => 101010 => [1,1,1,1,1,1] => 0
[1,2] => [1,0,1,1,0,0] => 101100 => [1,1,2,2] => 0
[2,1] => [1,1,0,0,1,0] => 110010 => [2,2,1,1] => 0
[3] => [1,1,1,0,0,0] => 111000 => [3,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 10101010 => [1,1,1,1,1,1,1,1] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => 10101100 => [1,1,1,1,2,2] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => 10110010 => [1,1,2,2,1,1] => 0
[1,3] => [1,0,1,1,1,0,0,0] => 10111000 => [1,1,3,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => 11001010 => [2,2,1,1,1,1] => 0
[2,2] => [1,1,0,0,1,1,0,0] => 11001100 => [2,2,2,2] => 0
[3,1] => [1,1,1,0,0,0,1,0] => 11100010 => [3,3,1,1] => 0
[4] => [1,1,1,1,0,0,0,0] => 11110000 => [4,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => [1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => [1,1,1,1,1,1,2,2] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 1010110010 => [1,1,1,1,2,2,1,1] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => [1,1,1,1,3,3] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 1011001010 => [1,1,2,2,1,1,1,1] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 1011001100 => [1,1,2,2,2,2] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => [1,1,3,3,1,1] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 1011110000 => [1,1,4,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 1100101010 => [2,2,1,1,1,1,1,1] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 1100101100 => [2,2,1,1,2,2] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 1100110010 => [2,2,2,2,1,1] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 1100111000 => [2,2,3,3] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 1110001010 => [3,3,1,1,1,1] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 1110001100 => [3,3,2,2] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => [4,4,1,1] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => [5,5] => 0
[1,1,1,1,1,1] => [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] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => [1,1,1,1,1,1,1,1,2,2] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => [1,1,1,1,1,1,2,2,1,1] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => [1,1,1,1,1,1,3,3] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => [1,1,1,1,2,2,1,1,1,1] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => [1,1,1,1,2,2,2,2] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => [1,1,1,1,3,3,1,1] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => [1,1,1,1,4,4] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => [1,1,2,2,1,1,1,1,1,1] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => [1,1,2,2,1,1,2,2] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => [1,1,2,2,2,2,1,1] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => [1,1,2,2,3,3] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => [1,1,3,3,1,1,1,1] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => [1,1,3,3,2,2] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => [1,1,4,4,1,1] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => [1,1,5,5] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => [2,2,1,1,1,1,1,1,1,1] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => [2,2,1,1,1,1,2,2] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => [2,2,1,1,2,2,1,1] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => [2,2,1,1,3,3] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => [2,2,2,2,1,1,1,1] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => [2,2,2,2,2,2] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => [2,2,3,3,1,1] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => [2,2,4,4] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => [3,3,1,1,1,1,1,1] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => [3,3,1,1,2,2] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => [3,3,2,2,1,1] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => [3,3,3,3] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => [4,4,1,1,1,1] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => [4,4,2,2] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => [5,5,1,1] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => [6,6] => 0
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Description
The sum of the heights of the valleys of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
bounce path
Description
The bounce path determined by 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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