Your data matches 58 different statistics following compositions of up to 3 maps.
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Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 2
Description
The number of up steps after the last double rise of a Dyck path.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00102: Dyck paths rise compositionInteger compositions
Mp00041: Integer compositions conjugateInteger compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1] => [2] => 2
[2,1] => [1,1,0,0]
=> [2] => [1,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => [3] => 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => [1,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1] => [2,1] => 2
[3,1,2] => [1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => 2
Description
The first part of an integer composition.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00109: Permutations descent wordBinary words
St000297: Binary words ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [2,1] => 1 => 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [1,2] => 0 => 0 = 1 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 11 => 2 = 3 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 01 => 0 = 1 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 10 => 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 10 => 1 = 2 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 00 => 0 = 1 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 011 => 0 = 1 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 101 => 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 101 => 1 = 2 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 0 = 1 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 0 = 1 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 110 => 2 = 3 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 010 => 0 = 1 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 110 => 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 010 => 0 = 1 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 010 => 0 = 1 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 100 => 1 = 2 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 100 => 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 100 => 1 = 2 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1111 => 4 = 5 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 0111 => 0 = 1 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1011 => 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 1011 => 1 = 2 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0011 => 0 = 1 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 0011 => 0 = 1 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1101 => 2 = 3 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0101 => 0 = 1 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 1101 => 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1101 => 2 = 3 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0101 => 0 = 1 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0101 => 0 = 1 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1001 => 1 = 2 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1001 => 1 = 2 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1001 => 1 = 2 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1001 => 1 = 2 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1001 => 1 = 2 - 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1001 => 1 = 2 - 1
[1,4,6,8,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ? => ? = 1 - 1
[1,6,5,4,8,7,3,2] => [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,6,2,3,4,7,8,1] => ? => ? = 1 - 1
[1,4,6,5,8,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,2,7,8,1] => ? => ? = 1 - 1
[1,4,6,7,5,8,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,7,8,1] => ? => ? = 3 - 1
[1,3,5,6,8,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ? => ? = 1 - 1
[1,4,5,3,8,7,6,2] => [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => ? => ? = 1 - 1
[1,4,6,5,3,8,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,2,5,8,1] => ? => ? = 1 - 1
[1,3,6,5,4,8,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ? => ? = 1 - 1
[1,5,4,3,6,8,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [6,7,5,2,3,4,8,1] => ? => ? = 1 - 1
[1,4,6,5,7,3,8,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ? => ? = 3 - 1
[1,3,4,6,7,5,8,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ? => ? = 3 - 1
[2,1,5,7,6,8,4,3] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,1,2] => ? => ? = 2 - 1
[2,1,4,6,7,8,5,3] => [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,3,8,1,2] => ? => ? = 3 - 1
[2,1,5,4,7,8,6,3] => [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,1,2] => ? => ? = 2 - 1
[2,1,6,5,4,8,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,1,2] => ? => ? = 1 - 1
[2,1,5,6,4,8,7,3] => [1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,1,2] => ? => ? = 1 - 1
[2,1,5,4,6,8,7,3] => [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ? => ? = 1 - 1
[2,1,4,5,6,8,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,3,8,1,2] => ? => ? = 1 - 1
[2,1,6,7,5,4,8,3] => [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,8,1,2] => ? => ? = 3 - 1
[2,1,5,7,6,4,8,3] => [1,1,0,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,1,2] => ? => ? = 2 - 1
[2,1,4,7,6,5,8,3] => [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ? => ? = 2 - 1
[2,1,4,6,7,5,8,3] => [1,1,0,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,8,1,2] => ? => ? = 3 - 1
[1,2,6,5,8,7,4,3] => [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,7,8,2,1] => ? => ? = 1 - 1
[1,2,5,6,8,7,4,3] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => ? => ? = 1 - 1
[1,2,5,7,6,8,4,3] => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,2,1] => ? => ? = 2 - 1
[1,2,4,6,8,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,8,2,1] => ? => ? = 1 - 1
[1,2,5,4,8,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,2,1] => ? => ? = 1 - 1
[1,2,5,4,7,8,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => ? => ? = 2 - 1
[1,2,6,5,4,8,7,3] => [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => ? => ? = 1 - 1
[1,2,5,4,6,8,7,3] => [1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,2,1] => ? => ? = 1 - 1
[1,2,5,7,6,4,8,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => ? => ? = 2 - 1
[1,2,5,4,7,6,8,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,2,1] => ? => ? = 2 - 1
[1,3,2,6,8,7,5,4] => [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,2,3,1] => ? => ? = 1 - 1
[1,3,2,7,6,8,5,4] => [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,3,1] => ? => ? = 2 - 1
[1,3,2,6,7,8,5,4] => [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,8,2,3,1] => ? => ? = 3 - 1
[1,3,2,5,8,7,6,4] => [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ? => ? = 1 - 1
[1,3,2,5,6,8,7,4] => [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,3,1] => ? => ? = 1 - 1
[1,3,2,7,6,5,8,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,2,3,1] => ? => ? = 2 - 1
[1,3,2,6,7,5,8,4] => [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,2,3,1] => ? => ? = 3 - 1
[1,3,2,6,5,7,8,4] => [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,2,3,1] => ? => ? = 3 - 1
[1,4,3,2,7,6,8,5] => [1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,2,3,4,1] => ? => ? = 2 - 1
[1,3,4,2,7,8,6,5] => [1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,2,4,1] => ? => ? = 2 - 1
[1,3,4,2,7,6,8,5] => [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => ? => ? = 2 - 1
[1,2,4,3,7,6,8,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,4,2,1] => ? => ? = 2 - 1
[1,4,5,3,2,7,8,6] => [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,2,4,5,1] => ? => ? = 2 - 1
[2,1,5,4,3,8,7,6] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,1,2] => ? => ? = 1 - 1
[1,3,2,5,4,8,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,2,3,1] => ? => ? = 1 - 1
[2,1,3,5,4,7,8,6] => [1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [7,6,8,4,5,3,1,2] => ? => ? = 2 - 1
[1,2,3,5,4,8,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ? => ? = 1 - 1
[1,2,4,3,5,8,7,6] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => ? => ? = 1 - 1
Description
The number of leading ones in a binary word.
Mp00066: Permutations inversePermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00315: Integer compositions inverse Foata bijectionInteger compositions
St000383: Integer compositions ⟶ ℤResult quality: 88% values known / values provided: 93%distinct values known / distinct values provided: 88%
Values
[1,2] => [1,2] => [2] => [2] => 2
[2,1] => [2,1] => [1,1] => [1,1] => 1
[1,2,3] => [1,2,3] => [3] => [3] => 3
[1,3,2] => [1,3,2] => [2,1] => [2,1] => 1
[2,1,3] => [2,1,3] => [1,2] => [1,2] => 2
[2,3,1] => [3,1,2] => [1,2] => [1,2] => 2
[3,1,2] => [2,3,1] => [2,1] => [2,1] => 1
[3,2,1] => [3,2,1] => [1,1,1] => [1,1,1] => 1
[1,2,3,4] => [1,2,3,4] => [4] => [4] => 4
[1,2,4,3] => [1,2,4,3] => [3,1] => [3,1] => 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [2,2] => 2
[1,3,4,2] => [1,4,2,3] => [2,2] => [2,2] => 2
[1,4,2,3] => [1,3,4,2] => [3,1] => [3,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => [1,2,1] => 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,3] => 3
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [2,1,1] => 1
[2,3,1,4] => [3,1,2,4] => [1,3] => [1,3] => 3
[2,3,4,1] => [4,1,2,3] => [1,3] => [1,3] => 3
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [2,1,1] => 1
[2,4,3,1] => [4,1,3,2] => [1,2,1] => [2,1,1] => 1
[3,1,2,4] => [2,3,1,4] => [2,2] => [2,2] => 2
[3,1,4,2] => [2,4,1,3] => [2,2] => [2,2] => 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => [1,1,2] => 2
[3,2,4,1] => [4,2,1,3] => [1,1,2] => [1,1,2] => 2
[3,4,1,2] => [3,4,1,2] => [2,2] => [2,2] => 2
[3,4,2,1] => [4,3,1,2] => [1,1,2] => [1,1,2] => 2
[4,1,2,3] => [2,3,4,1] => [3,1] => [3,1] => 1
[4,1,3,2] => [2,4,3,1] => [2,1,1] => [1,2,1] => 1
[4,2,1,3] => [3,2,4,1] => [1,2,1] => [2,1,1] => 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => [2,1,1] => 1
[4,3,1,2] => [3,4,2,1] => [2,1,1] => [1,2,1] => 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [5] => 5
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [4,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [3,2] => 2
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [3,2] => 2
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => [4,1] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => [1,3,1] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [2,3] => 3
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [2,2,1] => 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [2,3] => 3
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [2,3] => 3
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => [2,2,1] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => [2,2,1] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => [3,2] => 2
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => [3,2] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => [2,1,2] => 2
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => [2,1,2] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => [3,2] => 2
[1,4,5,3,2] => [1,5,4,2,3] => [2,1,2] => [2,1,2] => 2
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [8] => [8] => ? = 8
[1,6,7,8,5,4,3,2] => [1,8,7,6,5,2,3,4] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,6,7,5,8,4,3,2] => [1,8,7,6,4,2,3,5] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,6,5,7,8,4,3,2] => [1,8,7,6,3,2,4,5] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,6,5,4,7,8,3,2] => [1,8,7,4,3,2,5,6] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,4,5,6,7,8,3,2] => [1,8,7,2,3,4,5,6] => [2,1,5] => [2,1,5] => ? = 5
[1,3,7,8,6,5,4,2] => [1,8,2,7,6,5,3,4] => ? => ? => ? = 2
[1,3,5,6,7,8,4,2] => [1,8,2,7,3,4,5,6] => [2,2,4] => [2,2,4] => ? = 4
[1,4,3,5,8,7,6,2] => [1,8,3,2,4,7,6,5] => ? => ? => ? = 1
[1,3,4,5,7,8,6,2] => [1,8,2,3,4,7,5,6] => [2,4,2] => [4,2,2] => ? = 2
[1,3,6,5,4,8,7,2] => [1,8,2,5,4,3,7,6] => ? => ? => ? = 1
[1,3,4,5,6,8,7,2] => [1,8,2,3,4,5,7,6] => [2,5,1] => [2,5,1] => ? = 1
[1,6,7,5,4,3,8,2] => [1,8,6,5,4,2,3,7] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,6,5,7,4,3,8,2] => [1,8,6,5,3,2,4,7] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,4,5,6,7,3,8,2] => [1,8,6,2,3,4,5,7] => [2,1,5] => [2,1,5] => ? = 5
[1,3,5,6,7,4,8,2] => [1,8,2,6,3,4,5,7] => [2,2,4] => [2,2,4] => ? = 4
[1,3,4,5,7,6,8,2] => [1,8,2,3,4,6,5,7] => [2,4,2] => [4,2,2] => ? = 2
[1,6,5,4,3,7,8,2] => [1,8,5,4,3,2,6,7] => [2,1,1,1,3] => [1,1,2,1,3] => ? = 3
[1,3,5,6,4,7,8,2] => [1,8,2,5,3,4,6,7] => [2,2,4] => [2,2,4] => ? = 4
[1,4,5,3,6,7,8,2] => [1,8,4,2,3,5,6,7] => [2,1,5] => [2,1,5] => ? = 5
[1,3,5,4,6,7,8,2] => [1,8,2,4,3,5,6,7] => [2,2,4] => [2,2,4] => ? = 4
[1,4,3,5,6,7,8,2] => [1,8,3,2,4,5,6,7] => [2,1,5] => [2,1,5] => ? = 5
[1,3,4,5,6,7,8,2] => [1,8,2,3,4,5,6,7] => [2,6] => [2,6] => ? = 6
[2,1,6,7,8,5,4,3] => [2,1,8,7,6,3,4,5] => [1,2,1,1,3] => [1,2,1,1,3] => ? = 3
[2,1,6,7,5,8,4,3] => [2,1,8,7,5,3,4,6] => [1,2,1,1,3] => [1,2,1,1,3] => ? = 3
[2,1,6,5,7,8,4,3] => [2,1,8,7,4,3,5,6] => [1,2,1,1,3] => [1,2,1,1,3] => ? = 3
[2,1,5,6,7,8,4,3] => [2,1,8,7,3,4,5,6] => [1,2,1,4] => [2,1,1,4] => ? = 4
[2,1,4,5,6,8,7,3] => [2,1,8,3,4,5,7,6] => [1,2,4,1] => [2,4,1,1] => ? = 1
[2,1,6,7,5,4,8,3] => [2,1,8,6,5,3,4,7] => [1,2,1,1,3] => [1,2,1,1,3] => ? = 3
[2,1,6,5,7,4,8,3] => [2,1,8,6,4,3,5,7] => [1,2,1,1,3] => [1,2,1,1,3] => ? = 3
[2,1,5,6,7,4,8,3] => [2,1,8,6,3,4,5,7] => [1,2,1,4] => [2,1,1,4] => ? = 4
[2,1,5,6,4,7,8,3] => [2,1,8,5,3,4,6,7] => [1,2,1,4] => [2,1,1,4] => ? = 4
[2,1,5,4,6,7,8,3] => [2,1,8,4,3,5,6,7] => [1,2,1,4] => [2,1,1,4] => ? = 4
[2,1,4,5,6,7,8,3] => [2,1,8,3,4,5,6,7] => [1,2,5] => [1,2,5] => ? = 5
[1,2,6,5,8,7,4,3] => [1,2,8,7,4,3,6,5] => [3,1,1,2,1] => [3,1,1,2,1] => ? = 1
[1,2,5,6,7,8,4,3] => [1,2,8,7,3,4,5,6] => [3,1,4] => [3,1,4] => ? = 4
[1,2,6,5,4,8,7,3] => [1,2,8,5,4,3,7,6] => ? => ? => ? = 1
[1,2,5,7,6,4,8,3] => [1,2,8,6,3,5,4,7] => ? => ? => ? = 2
[1,2,5,6,7,4,8,3] => [1,2,8,6,3,4,5,7] => [3,1,4] => [3,1,4] => ? = 4
[1,2,5,4,6,7,8,3] => [1,2,8,4,3,5,6,7] => [3,1,4] => [3,1,4] => ? = 4
[1,2,4,5,6,7,8,3] => [1,2,8,3,4,5,6,7] => [3,5] => [3,5] => ? = 5
[1,3,2,5,6,7,8,4] => [1,3,2,8,4,5,6,7] => [2,2,4] => [2,2,4] => ? = 4
[2,1,3,6,8,7,5,4] => [2,1,3,8,7,4,6,5] => [1,3,1,2,1] => [3,1,2,1,1] => ? = 1
[2,1,3,6,5,8,7,4] => [2,1,3,8,5,4,7,6] => [1,3,1,2,1] => [3,1,2,1,1] => ? = 1
[2,1,3,5,6,7,8,4] => [2,1,3,8,4,5,6,7] => [1,3,4] => [1,3,4] => ? = 4
[1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => [4,1,1,1,1] => [1,1,1,4,1] => ? = 1
[1,2,3,7,8,6,5,4] => [1,2,3,8,7,6,4,5] => [4,1,1,2] => [1,1,4,2] => ? = 2
[1,2,3,7,6,8,5,4] => [1,2,3,8,7,5,4,6] => [4,1,1,2] => [1,1,4,2] => ? = 2
[1,2,3,7,6,5,8,4] => [1,2,3,8,6,5,4,7] => [4,1,1,2] => [1,1,4,2] => ? = 2
[2,3,1,4,8,7,6,5] => [3,1,2,4,8,7,6,5] => [1,4,1,1,1] => [1,1,4,1,1] => ? = 1
Description
The last part of an integer composition.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2
[2,1] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => [[1,3],[2]]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[1,4],[2],[3]]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2
[1,4,6,8,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,2,7,8,1] => ?
=> ? = 1
[1,6,5,4,8,7,3,2] => [1,0,1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,6,2,3,4,7,8,1] => ?
=> ? = 1
[1,4,6,5,8,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,2,7,8,1] => ?
=> ? = 1
[1,4,6,7,5,8,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [6,4,3,5,2,7,8,1] => ?
=> ? = 3
[1,3,5,6,8,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ?
=> ? = 1
[1,4,3,7,6,8,5,2] => [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,2,3,8,1] => ?
=> ? = 2
[1,4,5,3,8,7,6,2] => [1,0,1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,2,4,8,1] => ?
=> ? = 1
[1,3,5,4,8,7,6,2] => [1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ? = 1
[1,4,6,5,3,8,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,2,5,8,1] => ?
=> ? = 1
[1,3,6,5,4,8,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ? = 1
[1,5,4,3,6,8,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [6,7,5,2,3,4,8,1] => ?
=> ? = 1
[1,6,5,7,4,3,8,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,3,5,6,8,1] => ?
=> ? = 3
[1,4,6,5,7,3,8,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,2,6,8,1] => ?
=> ? = 3
[1,3,4,6,7,5,8,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ?
=> ? = 3
[2,1,5,7,6,8,4,3] => [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,1,2] => ?
=> ? = 2
[2,1,6,5,7,8,4,3] => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,5,3,4,7,8,1,2] => ?
=> ? = 3
[2,1,4,7,8,6,5,3] => [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,8,1,2] => ?
=> ? = 2
[2,1,5,4,8,7,6,3] => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,1,2] => ?
=> ? = 1
[2,1,5,4,7,8,6,3] => [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,1,2] => ?
=> ? = 2
[2,1,4,5,7,8,6,3] => [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [6,5,7,4,3,8,1,2] => ?
=> ? = 2
[2,1,6,5,4,8,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,1,2] => ?
=> ? = 1
[2,1,5,6,4,8,7,3] => [1,1,0,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,8,1,2] => ?
=> ? = 1
[2,1,5,4,6,8,7,3] => [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,1,2] => ?
=> ? = 1
[2,1,4,5,6,8,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,3,8,1,2] => ?
=> ? = 1
[2,1,6,7,5,4,8,3] => [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,8,1,2] => ?
=> ? = 3
[2,1,4,7,6,5,8,3] => [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,3,8,1,2] => ?
=> ? = 2
[2,1,5,4,7,6,8,3] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ?
=> ? = 2
[1,2,5,8,7,6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,5,6,3,7,8,2,1] => ?
=> ? = 1
[1,2,6,5,8,7,4,3] => [1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [5,6,3,4,7,8,2,1] => ?
=> ? = 1
[1,2,5,6,8,7,4,3] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,8,2,1] => ?
=> ? = 1
[1,2,5,7,6,8,4,3] => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,4,5,3,7,8,2,1] => ?
=> ? = 2
[1,2,4,6,8,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,8,2,1] => ?
=> ? = 1
[1,2,5,4,8,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,8,2,1] => ?
=> ? = 1
[1,2,5,4,7,8,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [6,5,7,3,4,8,2,1] => ?
=> ? = 2
[1,2,6,5,4,8,7,3] => [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,8,2,1] => ?
=> ? = 1
[1,2,5,4,6,8,7,3] => [1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,8,2,1] => ?
=> ? = 1
[1,2,5,7,6,4,8,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,4,5,3,6,8,2,1] => ?
=> ? = 2
[1,2,5,4,7,6,8,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,2,1] => ?
=> ? = 2
[1,3,2,6,8,7,5,4] => [1,0,1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,8,2,3,1] => ?
=> ? = 1
[1,3,2,7,6,8,5,4] => [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [6,4,5,7,8,2,3,1] => ?
=> ? = 2
[1,3,2,6,7,8,5,4] => [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,8,2,3,1] => ?
=> ? = 3
[1,3,2,5,8,7,6,4] => [1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,8,2,3,1] => ?
=> ? = 1
[1,3,2,5,6,8,7,4] => [1,0,1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [6,7,5,4,8,2,3,1] => ?
=> ? = 1
[1,3,2,7,6,5,8,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [7,4,5,6,8,2,3,1] => ?
=> ? = 2
[1,3,2,6,7,5,8,4] => [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,8,2,3,1] => ?
=> ? = 3
[1,3,2,6,5,7,8,4] => [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,8,2,3,1] => ?
=> ? = 3
[1,4,3,2,7,6,8,5] => [1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,2,3,4,1] => ?
=> ? = 2
[1,3,4,2,7,8,6,5] => [1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,3,2,4,1] => ?
=> ? = 2
[1,3,4,2,7,6,8,5] => [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,2,4,1] => ?
=> ? = 2
[1,2,4,3,7,6,8,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,4,2,1] => ?
=> ? = 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Mp00066: Permutations inversePermutations
Mp00109: Permutations descent wordBinary words
Mp00104: Binary words reverseBinary words
St000326: Binary words ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0 => 2
[2,1] => [2,1] => 1 => 1 => 1
[1,2,3] => [1,2,3] => 00 => 00 => 3
[1,3,2] => [1,3,2] => 01 => 10 => 1
[2,1,3] => [2,1,3] => 10 => 01 => 2
[2,3,1] => [3,1,2] => 10 => 01 => 2
[3,1,2] => [2,3,1] => 01 => 10 => 1
[3,2,1] => [3,2,1] => 11 => 11 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 4
[1,2,4,3] => [1,2,4,3] => 001 => 100 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 010 => 2
[1,3,4,2] => [1,4,2,3] => 010 => 010 => 2
[1,4,2,3] => [1,3,4,2] => 001 => 100 => 1
[1,4,3,2] => [1,4,3,2] => 011 => 110 => 1
[2,1,3,4] => [2,1,3,4] => 100 => 001 => 3
[2,1,4,3] => [2,1,4,3] => 101 => 101 => 1
[2,3,1,4] => [3,1,2,4] => 100 => 001 => 3
[2,3,4,1] => [4,1,2,3] => 100 => 001 => 3
[2,4,1,3] => [3,1,4,2] => 101 => 101 => 1
[2,4,3,1] => [4,1,3,2] => 101 => 101 => 1
[3,1,2,4] => [2,3,1,4] => 010 => 010 => 2
[3,1,4,2] => [2,4,1,3] => 010 => 010 => 2
[3,2,1,4] => [3,2,1,4] => 110 => 011 => 2
[3,2,4,1] => [4,2,1,3] => 110 => 011 => 2
[3,4,1,2] => [3,4,1,2] => 010 => 010 => 2
[3,4,2,1] => [4,3,1,2] => 110 => 011 => 2
[4,1,2,3] => [2,3,4,1] => 001 => 100 => 1
[4,1,3,2] => [2,4,3,1] => 011 => 110 => 1
[4,2,1,3] => [3,2,4,1] => 101 => 101 => 1
[4,2,3,1] => [4,2,3,1] => 101 => 101 => 1
[4,3,1,2] => [3,4,2,1] => 011 => 110 => 1
[4,3,2,1] => [4,3,2,1] => 111 => 111 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 5
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1000 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 0100 => 2
[1,2,4,5,3] => [1,2,5,3,4] => 0010 => 0100 => 2
[1,2,5,3,4] => [1,2,4,5,3] => 0001 => 1000 => 1
[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1100 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 0010 => 3
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 1
[1,3,4,2,5] => [1,4,2,3,5] => 0100 => 0010 => 3
[1,3,4,5,2] => [1,5,2,3,4] => 0100 => 0010 => 3
[1,3,5,2,4] => [1,4,2,5,3] => 0101 => 1010 => 1
[1,3,5,4,2] => [1,5,2,4,3] => 0101 => 1010 => 1
[1,4,2,3,5] => [1,3,4,2,5] => 0010 => 0100 => 2
[1,4,2,5,3] => [1,3,5,2,4] => 0010 => 0100 => 2
[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 0110 => 2
[1,4,3,5,2] => [1,5,3,2,4] => 0110 => 0110 => 2
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 0100 => 2
[1,4,5,3,2] => [1,5,4,2,3] => 0110 => 0110 => 2
[1,4,6,5,8,7,3,2] => [1,8,7,2,4,3,6,5] => ? => ? => ? = 1
[1,3,7,8,6,5,4,2] => [1,8,2,7,6,5,3,4] => ? => ? => ? = 2
[1,4,3,6,8,7,5,2] => [1,8,3,2,7,4,6,5] => ? => ? => ? = 1
[1,4,3,6,7,8,5,2] => [1,8,3,2,7,4,5,6] => ? => ? => ? = 3
[1,3,4,6,7,8,5,2] => [1,8,2,3,7,4,5,6] => ? => ? => ? = 3
[1,4,3,5,8,7,6,2] => [1,8,3,2,4,7,6,5] => ? => ? => ? = 1
[1,4,6,5,3,8,7,2] => [1,8,5,2,4,3,7,6] => ? => ? => ? = 1
[1,3,6,5,4,8,7,2] => [1,8,2,5,4,3,7,6] => ? => ? => ? = 1
[2,1,4,5,6,8,7,3] => [2,1,8,3,4,5,7,6] => ? => ? => ? = 1
[2,1,6,7,5,4,8,3] => [2,1,8,6,5,3,4,7] => ? => ? => ? = 3
[2,1,5,7,6,4,8,3] => [2,1,8,6,3,5,4,7] => ? => ? => ? = 2
[2,1,5,6,7,4,8,3] => [2,1,8,6,3,4,5,7] => ? => ? => ? = 4
[2,1,4,7,6,5,8,3] => [2,1,8,3,6,5,4,7] => ? => ? => ? = 2
[2,1,5,4,6,7,8,3] => [2,1,8,4,3,5,6,7] => ? => ? => ? = 4
[1,2,6,5,7,8,4,3] => [1,2,8,7,4,3,5,6] => ? => ? => ? = 3
[1,2,4,5,7,8,6,3] => [1,2,8,3,4,7,5,6] => ? => ? => ? = 2
[1,2,6,5,4,8,7,3] => [1,2,8,5,4,3,7,6] => ? => ? => ? = 1
[1,2,5,7,6,4,8,3] => [1,2,8,6,3,5,4,7] => ? => ? => ? = 2
[1,3,2,5,8,7,6,4] => [1,3,2,8,4,7,6,5] => ? => ? => ? = 1
[1,4,3,2,6,8,7,5] => [1,4,3,2,8,5,7,6] => ? => ? => ? = 1
[1,5,4,3,2,7,8,6] => [1,5,4,3,2,8,6,7] => ? => ? => ? = 2
[2,1,4,5,3,8,7,6] => [2,1,5,3,4,8,7,6] => ? => ? => ? = 1
[2,1,3,5,4,7,8,6] => [2,1,3,5,4,8,6,7] => ? => ? => ? = 2
[1,3,6,5,4,2,8,7] => [1,6,2,5,4,3,8,7] => ? => ? => ? = 1
[1,3,2,5,6,4,8,7] => [1,3,2,6,4,5,8,7] => ? => ? => ? = 1
[2,1,3,6,5,4,8,7] => [2,1,3,6,5,4,8,7] => ? => ? => ? = 1
[2,1,5,4,3,6,8,7] => [2,1,5,4,3,6,8,7] => ? => ? => ? = 1
[1,5,6,7,4,3,2,8] => [1,7,6,5,2,3,4,8] => ? => ? => ? = 4
[1,4,7,6,5,3,2,8] => [1,7,6,2,5,4,3,8] => ? => ? => ? = 2
[1,4,6,7,5,3,2,8] => [1,7,6,2,5,3,4,8] => ? => ? => ? = 3
[1,3,6,7,5,4,2,8] => [1,7,2,6,5,3,4,8] => ? => ? => ? = 3
[2,1,4,6,7,5,3,8] => [2,1,7,3,6,4,5,8] => ? => ? => ? = 3
[2,1,5,4,6,7,3,8] => [2,1,7,4,3,5,6,8] => ? => ? => ? = 4
[1,2,6,7,5,4,3,8] => [1,2,7,6,5,3,4,8] => ? => ? => ? = 3
[1,2,5,7,6,4,3,8] => [1,2,7,6,3,5,4,8] => ? => ? => ? = 2
[2,1,3,6,7,5,4,8] => [2,1,3,7,6,4,5,8] => ? => ? => ? = 3
[1,4,3,2,6,7,5,8] => [1,4,3,2,7,5,6,8] => ? => ? => ? = 3
[2,1,4,3,6,7,5,8] => [2,1,4,3,7,5,6,8] => ? => ? => ? = 3
[1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ? => ? => ? = 3
[1,5,6,4,3,2,7,8] => [1,6,5,4,2,3,7,8] => ? => ? => ? = 4
[1,4,6,5,3,2,7,8] => [1,6,5,2,4,3,7,8] => ? => ? => ? = 3
[1,4,5,6,3,2,7,8] => [1,6,5,2,3,4,7,8] => ? => ? => ? = 5
[2,1,5,6,4,3,7,8] => [2,1,6,5,3,4,7,8] => ? => ? => ? = 4
[2,1,4,6,5,3,7,8] => [2,1,6,3,5,4,7,8] => ? => ? => ? = 3
[2,3,1,5,4,6,7,8] => [3,1,2,5,4,6,7,8] => ? => ? => ? = 4
[2,1,3,5,4,6,7,8] => [2,1,3,5,4,6,7,8] => ? => ? => ? = 4
[1,2,4,3,8,6,7,5] => [1,2,4,3,8,6,7,5] => ? => ? => ? = 1
[1,2,8,4,6,7,5,3] => [1,2,8,4,7,5,6,3] => ? => ? => ? = 1
[1,3,2,8,6,7,5,4] => [1,3,2,8,7,5,6,4] => ? => ? => ? = 1
[1,3,4,2,8,6,7,5] => [1,4,2,3,8,6,7,5] => ? => ? => ? = 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00296: Dyck paths Knuth-KrattenthalerDyck paths
Mp00123: Dyck paths Barnabei-Castronuovo involutionDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,6,7,8,5,4,3,2] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,6,7,5,8,4,3,2] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,6,5,7,8,4,3,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,5,6,7,8,4,3,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,4,6,7,8,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,5,4,7,8,6,3,2] => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,4,5,7,8,6,3,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,6,5,4,7,8,3,2] => [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,5,4,6,7,8,3,2] => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 4 + 1
[1,3,6,7,8,5,4,2] => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,3,5,7,8,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,3,6,7,5,8,4,2] => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3 + 1
[1,3,5,7,6,8,4,2] => [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[1,3,6,5,7,8,4,2] => [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,4,3,6,7,8,5,2] => [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,3,4,6,7,8,5,2] => [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,4,5,3,7,8,6,2] => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,4,3,5,7,8,6,2] => [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[1,3,4,5,7,8,6,2] => [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? = 2 + 1
[1,6,7,5,4,3,8,2] => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,6,5,7,4,3,8,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,4,6,7,5,3,8,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,5,6,4,7,3,8,2] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4 + 1
[1,4,6,5,7,3,8,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,5,4,6,7,3,8,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 4 + 1
[1,4,5,6,7,3,8,2] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,3,5,7,6,4,8,2] => [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,3,6,5,7,4,8,2] => [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 1
[1,3,5,6,7,4,8,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 4 + 1
[1,4,3,6,7,5,8,2] => [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3 + 1
[1,3,4,6,7,5,8,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,3,4,5,7,6,8,2] => [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 2 + 1
[1,6,5,4,3,7,8,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
[1,5,4,6,3,7,8,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 4 + 1
[1,3,5,6,4,7,8,2] => [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 4 + 1
[1,4,3,6,5,7,8,2] => [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,5,4,3,6,7,8,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,4,5,3,6,7,8,2] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 5 + 1
[1,3,5,4,6,7,8,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 4 + 1
[1,4,3,5,6,7,8,2] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 5 + 1
[2,1,6,7,8,5,4,3] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[2,1,5,6,7,8,4,3] => [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[2,1,4,5,7,8,6,3] => [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? = 2 + 1
[2,1,6,7,5,4,8,3] => [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[2,1,5,6,7,4,8,3] => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 4 + 1
[2,1,5,6,4,7,8,3] => [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 4 + 1
[2,1,5,4,6,7,8,3] => [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 4 + 1
[1,2,6,7,8,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,2,5,7,6,8,4,3] => [1,0,1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[1,2,6,5,7,8,4,3] => [1,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 86% values known / values provided: 86%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,6,5,8,7,4,3,2] => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[1,5,7,6,8,4,3,2] => [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,5,6,7,8,4,3,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,4,7,8,6,5,3,2] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[1,4,6,8,7,5,3,2] => [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 1
[1,4,6,7,8,5,3,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,5,4,7,8,6,3,2] => [1,0,1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 2
[1,4,5,7,8,6,3,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 2
[1,4,6,5,8,7,3,2] => [1,0,1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 1
[1,5,4,6,8,7,3,2] => [1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,4,7,6,5,8,3,2] => [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2
[1,4,6,7,5,8,3,2] => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[1,6,5,4,7,8,3,2] => [1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[1,5,4,6,7,8,3,2] => [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 4
[1,3,6,8,7,5,4,2] => [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 1
[1,3,6,7,8,5,4,2] => [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3
[1,3,5,6,8,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,3,6,7,5,8,4,2] => [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> ? = 3
[1,3,6,5,7,8,4,2] => [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[1,4,3,7,8,6,5,2] => [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 2
[1,4,3,6,8,7,5,2] => [1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[1,4,3,7,6,8,5,2] => [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[1,4,3,6,7,8,5,2] => [1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[1,3,4,7,6,8,5,2] => [1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[1,4,5,3,7,8,6,2] => [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 2
[1,4,3,5,7,8,6,2] => [1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2
[1,5,6,4,3,8,7,2] => [1,0,1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
=> ? = 1
[1,4,6,5,3,8,7,2] => [1,0,1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
[1,4,5,6,3,8,7,2] => [1,0,1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,5,4,3,6,8,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
[1,4,5,3,6,8,7,2] => [1,0,1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,6,5,7,4,3,8,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,4,6,7,5,3,8,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 3
[1,5,6,4,7,3,8,2] => [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4
[1,4,6,5,7,3,8,2] => [1,0,1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 3
[1,5,4,6,7,3,8,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 4
[1,4,5,6,7,3,8,2] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,3,6,5,7,4,8,2] => [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[1,3,5,6,7,4,8,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 4
[1,4,3,6,7,5,8,2] => [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
[1,3,4,7,6,5,8,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,6,5,4,3,7,8,2] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,5,4,6,3,7,8,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4
[1,4,3,6,5,7,8,2] => [1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 3
[1,3,4,6,5,7,8,2] => [1,0,1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 3
[1,5,4,3,6,7,8,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,3,5,4,6,7,8,2] => [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 4
[2,1,6,8,7,5,4,3] => [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 1
[2,1,6,7,8,5,4,3] => [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 3
[2,1,5,7,8,6,4,3] => [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 2
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St001050: Set partitions ⟶ ℤResult quality: 75% values known / values provided: 75%distinct values known / distinct values provided: 88%
Values
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 2
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[4,1,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[4,2,1,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[4,3,1,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2
[1,2,5,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 1
[1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3
[1,3,5,2,4] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,3,5,4,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2
[1,4,3,5,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[1,4,5,2,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[1,4,5,3,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2
[2,3,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 7
[2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 7
[1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[1,6,7,8,5,4,3,2] => [6,7,8,5,4,3,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[1,6,5,8,7,4,3,2] => [6,8,5,7,4,3,1,2] => ?
=> ?
=> ? = 1
[1,7,6,5,8,4,3,2] => [7,6,5,8,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[1,6,7,5,8,4,3,2] => [6,7,5,8,4,3,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 3
[1,5,7,6,8,4,3,2] => [7,5,6,8,4,3,1,2] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[1,6,5,7,8,4,3,2] => [6,5,7,8,4,3,1,2] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 3
[1,5,6,7,8,4,3,2] => [5,6,7,8,4,3,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[1,4,6,7,8,5,3,2] => [6,7,8,4,5,3,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[1,5,4,7,8,6,3,2] => [5,7,8,4,6,3,1,2] => ?
=> ?
=> ? = 2
[1,4,5,8,7,6,3,2] => [8,7,4,5,6,3,1,2] => ?
=> ?
=> ? = 1
[1,6,5,4,8,7,3,2] => [6,5,8,4,7,3,1,2] => ?
=> ?
=> ? = 1
[1,4,6,5,8,7,3,2] => [6,8,4,5,7,3,1,2] => ?
=> ?
=> ? = 1
[1,5,4,6,8,7,3,2] => [5,8,4,6,7,3,1,2] => ?
=> ?
=> ? = 1
[1,7,6,5,4,8,3,2] => [7,6,5,4,8,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[1,4,7,6,5,8,3,2] => [7,6,4,5,8,3,1,2] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6,7}}
=> ? = 2
[1,4,6,7,5,8,3,2] => [6,7,4,5,8,3,1,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,7},{6}}
=> ? = 3
[1,6,5,4,7,8,3,2] => [6,5,4,7,8,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5,6},{7}}
=> ? = 3
[1,5,4,6,7,8,3,2] => [5,4,6,7,8,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4,5},{6},{7}}
=> ? = 4
[1,4,5,6,7,8,3,2] => [4,5,6,7,8,3,1,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> {{1,2,3,8},{4},{5},{6},{7}}
=> ? = 5
[1,3,6,8,7,5,4,2] => [8,6,7,5,3,4,1,2] => ?
=> ?
=> ? = 1
[1,3,6,7,8,5,4,2] => [6,7,8,5,3,4,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[1,3,6,7,5,8,4,2] => [6,7,5,8,3,4,1,2] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,7},{6}}
=> ? = 3
[1,3,5,7,6,8,4,2] => [7,5,6,8,3,4,1,2] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[1,3,6,5,7,8,4,2] => [6,5,7,8,3,4,1,2] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6},{7}}
=> ? = 3
[1,3,5,6,7,8,4,2] => [5,6,7,8,3,4,1,2] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 4
[1,4,3,6,8,7,5,2] => [8,4,6,7,3,5,1,2] => ?
=> ?
=> ? = 1
[1,4,3,7,6,8,5,2] => [7,4,6,8,3,5,1,2] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> {{1,2,3,4,8},{5,6,7}}
=> ? = 2
[1,4,3,6,7,8,5,2] => [4,6,7,8,3,5,1,2] => [1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> {{1,2,3,5,8},{4},{6},{7}}
=> ? = 3
[1,3,4,7,6,8,5,2] => [7,6,8,3,4,5,1,2] => ?
=> ?
=> ? = 2
[1,3,4,6,7,8,5,2] => [6,7,8,3,4,5,1,2] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 3
[1,4,5,3,7,8,6,2] => [4,5,7,8,3,6,1,2] => ?
=> ?
=> ? = 2
[1,3,5,4,8,7,6,2] => [8,5,7,3,4,6,1,2] => ?
=> ?
=> ? = 1
[1,3,5,4,7,8,6,2] => [5,7,8,3,4,6,1,2] => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 2
[1,4,3,5,8,7,6,2] => [8,4,7,3,5,6,1,2] => ?
=> ?
=> ? = 1
[1,4,3,5,7,8,6,2] => [4,7,8,3,5,6,1,2] => [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 2
[1,6,5,4,3,8,7,2] => [6,5,4,8,3,7,1,2] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1
[1,5,6,4,3,8,7,2] => [5,6,4,8,3,7,1,2] => [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,6},{5}}
=> ? = 1
[1,4,6,5,3,8,7,2] => [6,4,5,8,3,7,1,2] => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 1
[1,4,5,6,3,8,7,2] => [4,5,6,8,3,7,1,2] => ?
=> ?
=> ? = 1
[1,3,6,5,4,8,7,2] => [6,5,8,3,4,7,1,2] => ?
=> ?
=> ? = 1
[1,4,3,6,5,8,7,2] => [4,6,8,3,5,7,1,2] => [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 1
[1,4,5,3,6,8,7,2] => [4,5,8,3,6,7,1,2] => [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 1
[1,7,6,5,4,3,8,2] => [7,6,5,4,3,8,1,2] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,6,7}}
=> ? = 2
[1,6,7,5,4,3,8,2] => [6,7,5,4,3,8,1,2] => ?
=> ?
=> ? = 3
[1,6,5,7,4,3,8,2] => [6,5,7,4,3,8,1,2] => [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,7},{5,6}}
=> ? = 3
[1,4,6,7,5,3,8,2] => [6,7,4,5,3,8,1,2] => [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> {{1,2,8},{3,4,5,7},{6}}
=> ? = 3
[1,5,6,4,7,3,8,2] => [5,6,4,7,3,8,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> {{1,2,8},{3,7},{4,6},{5}}
=> ? = 4
Description
The number of terminal closers of a set partition. A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000759
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000759: Integer partitions ⟶ ℤResult quality: 71% values known / values provided: 71%distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1]
=> 2
[2,1] => [1,1,0,0]
=> []
=> []
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [2,1]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,2,1]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [3,2]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,1,1]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> 2
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2,2,1]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2,2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [2,1]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [4,3,2,1]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [4,3,1,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [4,2,2,1]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [4,2,2]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [4,2,1,1]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,2,1]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 2
[1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> [6,5,4,2,1,1]
=> ? = 3
[1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> [6,5,4,2,1]
=> ? = 3
[1,2,3,5,7,4,6] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [6,5,4,2]
=> ? = 1
[1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [6,5,4,2]
=> ? = 1
[1,2,3,6,4,7,5] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> [6,5,4,1,1]
=> ? = 2
[1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> [6,5,4,1,1]
=> ? = 2
[1,2,3,6,7,4,5] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> [6,5,4,1]
=> ? = 2
[1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> [6,5,4,1]
=> ? = 2
[1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> [6,5,3,3,1]
=> ? = 2
[1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> [6,5,3,2,2,1]
=> ? = 4
[1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> [6,5,3,2,2]
=> ? = 1
[1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> [6,5,3,2,1,1]
=> ? = 4
[1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> [6,5,3,2,1]
=> ? = 4
[1,2,4,5,7,3,6] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> [6,5,3,2]
=> ? = 1
[1,2,4,5,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> [6,5,3,2]
=> ? = 1
[1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> [6,5,3,1,1,1]
=> ? = 2
[1,2,4,6,3,7,5] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> [6,5,3,1,1]
=> ? = 2
[1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,2,1]
=> [6,5,3,1,1,1]
=> ? = 2
[1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,3,2,2,1]
=> [6,5,3,1,1]
=> ? = 2
[1,2,4,6,7,3,5] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> [6,5,3,1]
=> ? = 2
[1,2,4,6,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> [6,5,3,1]
=> ? = 2
[1,2,4,7,3,5,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[1,2,4,7,3,6,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[1,2,4,7,5,3,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[1,2,4,7,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[1,2,4,7,6,3,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [6,5,3]
=> ? = 1
[1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> [6,5,2,2,1,1]
=> ? = 3
[1,2,5,3,6,7,4] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> [6,5,2,2,1]
=> ? = 3
[1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> [6,5,2,2]
=> ? = 1
[1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> [6,5,2,2]
=> ? = 1
[1,2,5,4,6,3,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,2,2,1]
=> [6,5,2,2,1,1]
=> ? = 3
[1,2,5,4,6,7,3] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> [6,5,2,2,1]
=> ? = 3
[1,2,5,4,7,3,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> [6,5,2,2]
=> ? = 1
[1,2,5,4,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,4,2,2,2,1]
=> [6,5,2,2]
=> ? = 1
[1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> [6,5,2,1,1,1]
=> ? = 3
[1,2,5,6,3,7,4] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> ? = 3
[1,2,5,6,4,3,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,2,2,1]
=> [6,5,2,1,1,1]
=> ? = 3
[1,2,5,6,4,7,3] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,2,2,1]
=> [6,5,2,1,1]
=> ? = 3
[1,2,5,6,7,3,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [6,5,2,1]
=> ? = 3
[1,2,5,6,7,4,3] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,2,1]
=> [6,5,2,1]
=> ? = 3
[1,2,5,7,3,4,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[1,2,5,7,3,6,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[1,2,5,7,4,3,6] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[1,2,5,7,4,6,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[1,2,5,7,6,3,4] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[1,2,5,7,6,4,3] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [6,5,2]
=> ? = 1
[1,2,6,3,4,7,5] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> [6,5,1,1,1]
=> ? = 2
[1,2,6,3,5,7,4] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> [6,5,1,1,1]
=> ? = 2
[1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,2,2,2,1]
=> [6,5,1,1]
=> ? = 2
Description
The smallest missing part in an integer partition. In [3], this is referred to as the mex, the minimal excluded part of the partition. For compositions, this is studied in [sec.3.2., 1].
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000675The number of centered multitunnels of a Dyck path. St000025The number of initial rises of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000234The number of global ascents of a permutation. St000717The number of ordinal summands of a poset. St000237The number of small exceedances. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000068The number of minimal elements in a poset. St000989The number of final rises of a permutation. St000993The multiplicity of the largest part of an integer partition. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000843The decomposition number of a perfect matching. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000740The last entry of a permutation. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.