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Your data matches 68 different statistics following compositions of up to 3 maps.
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Mp00031: Dyck paths to 312-avoiding permutationPermutations
St000018: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 4
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 4
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 5
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 6
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 7
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions (i,i+1) needed to write π. Thus, it is also the Coxeter length of π.
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00109: Permutations descent wordBinary words
St000391: Binary words ⟶ ℤResult quality: 97% values known / values provided: 98%distinct values known / distinct values provided: 97%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 10 => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 01 => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 11 => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => 010 => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 110 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => 010 => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => 010 => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => 001 => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => 101 => 4
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 110 => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 101 => 4
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 011 => 5
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 111 => 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 1000 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => 0100 => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 1100 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => 0100 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => 0100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => 0010 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => 1010 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => 1100 => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => 1010 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => 0110 => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => 0100 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => 0100 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => 0010 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => 1010 => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => 0100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => 0010 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => 0010 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => 0001 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => 1001 => 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => 1010 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => 1001 => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => 0101 => 6
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => 1101 => 7
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1100 => 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => 1111111110 => ? = 45
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => 1111111110 => ? = 45
[]
=> [] => [] => ? => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [11,1,2,3,4,5,6,7,8,9,10] => 1000000000 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => 1000000000 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,4,6,8,10,12,1,3,5,7,9,11] => 00000100000 => ? = 6
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,12,1,11] => 11111111010 => ? = 46
Description
The sum of the positions of the ones in a binary word.
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 97%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 4
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 5
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [[1,3],[2,5],[4]]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [[1,2,4,5],[3]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [[1,3,5],[2],[4]]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [[1,3,4],[2],[5]]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 7
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [8,9,7,6,5,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 27
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 45
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? = 26
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,7,5,4,3,2,1,9] => [8,6,7,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? = 26
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [8,7,9,6,5,4,3,1,2] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 26
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [9,7,8,6,5,4,3,1,2] => [[1,3],[2,9],[4],[5],[6],[7],[8]]
=> ? = 26
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,8,7,6,5,4,3,2] => [9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? = 35
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 45
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8],[2],[9]]
=> ? = 9
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,9,1] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,10,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 10
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,12,1,11] => [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 46
Description
The (standard) major index of a standard tableau. A descent of a standard tableau T is an index i such that i+1 appears in a row strictly below the row of i. The (standard) major index is the the sum of the descents.
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
Mp00071: Permutations descent compositionInteger compositions
St000008: Integer compositions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 94%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,3] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [2,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,2] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [3,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [1,2,1] => 4
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,2] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,2,1] => 4
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [2,1,1] => 5
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [2,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [2,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [1,2,2] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [1,2,2] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => [2,1,2] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [2,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [3,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [1,2,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [3,2] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,3,1] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,2,2] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,3,1] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => [2,2,1] => 6
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,2,1] => 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [1,1,1,1,1,1,1,2] => ? = 28
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => [9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,2] => ? = 28
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,6,5,4,3,2,1,10] => [9,8,7,6,5,4,3,2,1,10] => [1,1,1,1,1,1,1,1,2] => ? = 36
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,6,5,4,3,2,1,9] => [7,8,6,5,4,3,2,1,9] => [2,1,1,1,1,1,2] => ? = 27
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,7,6,5,4,3,2] => [8,9,7,6,5,4,3,1,2] => [2,1,1,1,1,1,2] => ? = 27
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => [10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,2] => ? = 36
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [1,1,1,1,1,1,1,1,1,2] => ? = 45
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1,9] => [1,2,1,1,1,1,2] => ? = 26
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,7,5,4,3,2,1,9] => [8,6,7,5,4,3,2,1,9] => [1,2,1,1,1,1,2] => ? = 26
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,7,9,6,5,4,3,2] => [8,7,9,6,5,4,3,1,2] => [1,2,1,1,1,1,2] => ? = 26
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,8,6,5,4,3,2] => [9,7,8,6,5,4,3,1,2] => [1,2,1,1,1,1,2] => ? = 26
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [11,10,9,8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,1,1,1,2] => ? = 45
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => [1,1,1,1,1,1,3] => ? = 21
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,9,10] => [1,1,1,1,1,1,1,3] => ? = 28
[]
=> [] => [] => [] => ? = 0
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,9,1] => [2,6,1] => ? = 10
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,1] => ? = 36
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [9] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [10] => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => [1,1,7] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => [1,1,1,6] => ? = 6
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => [1,1,1,1,5] => ? = 10
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => [1,1,1,1,1,4] => ? = 15
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => [1,1,8] => ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7,8,9,10] => [4,3,2,1,5,6,7,8,9,10] => [1,1,1,7] => ? = 6
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => [1,1,1,1,6] => ? = 10
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1,7,8,9,10] => [6,5,4,3,2,1,7,8,9,10] => [1,1,1,1,1,5] => ? = 15
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9,10] => [7,6,5,4,3,2,1,8,9,10] => [1,1,1,1,1,1,4] => ? = 21
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8,9] => [6,3] => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1,9] => [7,2] => ? = 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,1,10] => [2,3,4,5,6,7,8,9,1,10] => [8,2] => ? = 8
Description
The major index of the composition. The descents of a composition [c1,c2,,ck] are the partial sums c1,c1+c2,,c1++ck1, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000246: Permutations ⟶ ℤResult quality: 81% values known / values provided: 81%distinct values known / distinct values provided: 94%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 4
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 4
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 5
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 6
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 5
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 6
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 6
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1,3] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ? = 3
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,1,4] => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,1,5] => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ? = 5
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? = 7
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,3,7] => ? = 8
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => ? = 9
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 10
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,2,3,7] => ? = 9
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1,2,4,7] => ? = 10
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,3,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,8,2,1] => ? = 12
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,5,2,3,1] => ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [6,7,4,5,2,3,8,1] => ? = 9
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,2,3,8,1] => ? = 13
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [6,7,2,3,4,5,8,1] => ? = 13
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,3,4,1,2] => ? = 6
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,4,1,2] => ? = 6
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,6,1,2] => ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,4,6,1,2] => ? = 6
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,8,1,2] => ? = 8
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,8,1,2] => ? = 10
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [8,7,5,4,3,2,1,6] => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,3,2,1,6] => ? = 6
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,1,8] => ? = 12
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,2,3,1,4] => ? = 6
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [6,7,5,8,2,3,1,4] => ? = 8
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,1,8] => ? = 20
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [4,5,6,7,8,1,2,3] => ? = 13
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,1,2,4] => ? = 6
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [7,5,6,8,3,1,2,4] => ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,1,2,8] => ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,1,2,5] => ? = 9
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,4,1,2,6] => ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [7,5,6,3,4,1,2,8] => ? = 10
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,1,2,8] => ? = 12
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [7,5,3,4,6,1,2,8] => ? = 12
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => ? = 13
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,2,1,3,8] => ? = 9
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,2,1,4,8] => ? = 10
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,1,7,8] => ? = 20
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,1,2,3,8] => ? = 10
Description
The number of non-inversions of a permutation. For a permutation of {1,,n}, this is given by \operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi).
Matching statistic: St001161
Mp00030: Dyck paths zeta mapDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St001161: Dyck paths ⟶ ℤResult quality: 78% values known / values provided: 80%distinct values known / distinct values provided: 78%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[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]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 27
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 27
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 45
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 35
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 35
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 45
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 21
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 29
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 29
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 28
[]
=> []
=> []
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 10
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 36
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
Description
The major index north count of a Dyck path. The descent set \operatorname{des}(D) of a Dyck path D = D_1 \cdots D_{2n} with D_i \in \{N,E\} is given by all indices i such that D_i = E and D_{i+1} = N. This is, the positions of the valleys of D. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, \sum_{i \in \operatorname{des}(D)} i, see [[St000027]]. The '''major index north count''' is given by \sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}.
Mp00030: Dyck paths zeta mapDyck paths
Mp00099: Dyck paths bounce pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St000012: Dyck paths ⟶ ℤResult quality: 66% values known / values provided: 80%distinct values known / distinct values provided: 66%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 20
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 10
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 9
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 10
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 20
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 10
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 14
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 16
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 16
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 18
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 20
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 22
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 24
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 22
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 24
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 36
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 27
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 27
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 36
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 45
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 35
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 35
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 45
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 21
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 29
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 29
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 28
[]
=> []
=> []
=> []
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 9
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 10
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 36
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
Description
The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' (a_1,\ldots,a_n) such that a_1 = 0, a_{k+1} \leq a_k + 1. 2. The generating function \mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)} satisfy the recurrence \mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q). 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of q,t-Catalan numbers.
Mp00030: Dyck paths zeta mapDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St000947: Dyck paths ⟶ ℤResult quality: 78% values known / values provided: 80%distinct values known / distinct values provided: 78%
Values
[1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[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]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 12
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 9
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 10
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 10
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 16
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 18
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 28
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 27
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 27
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 36
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 45
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 35
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 26
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 35
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 45
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 21
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 29
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 29
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 28
[]
=> []
=> []
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 9
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 10
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 36
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The major index east count of a Dyck path. The descent set \operatorname{des}(D) of a Dyck path D = D_1 \cdots D_{2n} with D_i \in \{N,E\} is given by all indices i such that D_i = E and D_{i+1} = N. This is, the positions of the valleys of D. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, \sum_{i \in \operatorname{des}(D)} i, see [[St000027]]. The '''major index east count''' is given by \sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}.
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 79% values known / values provided: 79%distinct values known / distinct values provided: 81%
Values
[1,0]
=> [1,0]
=> []
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 4
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 5
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 7
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1,1]
=> ? = 11
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,3,3,1]
=> ? = 11
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,3,2,1]
=> ? = 11
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,2,1]
=> ? = 12
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1]
=> ? = 13
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2]
=> ? = 10
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [5,5,2,1]
=> ? = 9
[1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [5,4,4,1]
=> ? = 10
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4]
=> ? = 12
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,1]
=> ? = 6
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,2,1,1,1]
=> ? = 9
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,3,2,1]
=> ? = 13
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2,1]
=> ? = 18
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,2,1,1]
=> ? = 13
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,2]
=> ? = 18
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,1,1]
=> ? = 19
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,2,1]
=> ? = 18
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1,1]
=> ? = 19
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 20
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,1,1]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,1,1,1]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,1,1]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? = 8
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,2,1,1]
=> ? = 6
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,1,1,1]
=> ? = 6
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,1]
=> ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? = 8
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,1]
=> ? = 10
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,4,2,2,2,2]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,4,4,1,1,1,1]
=> ? = 6
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ? = 12
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,3,2,2,1,1]
=> ? = 6
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,1,1,1]
=> ? = 8
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,1,1]
=> ? = 8
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,1,1]
=> ? = 20
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,2,2,2,1]
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,2,1,1]
=> ? = 9
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,2,2,1,1]
=> ? = 6
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1,1,1]
=> ? = 8
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,1,1]
=> ? = 8
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [5,5,5,1]
=> ? = 8
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2,2,1]
=> ? = 9
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,1,1,1]
=> ? = 8
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1,1,1]
=> ? = 9
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,1,1]
=> ? = 12
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1]
=> ? = 12
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2,1]
=> ? = 13
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2]
=> ? = 10
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [5,5,5,2]
=> ? = 10
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,2,2,1]
=> ? = 20
Description
The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells c in the diagram of an integer partition \lambda for which \operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}. See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].
Mp00029: Dyck paths to binary tree: left tree, up step, right tree, down stepBinary trees
St000161: Binary trees ⟶ ℤResult quality: 78% values known / values provided: 78%distinct values known / distinct values provided: 84%
Values
[1,0]
=> [.,.]
=> 0
[1,0,1,0]
=> [[.,.],.]
=> 0
[1,1,0,0]
=> [.,[.,.]]
=> 1
[1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 2
[1,1,1,0,0,0]
=> [.,[.,[.,.]]]
=> 3
[1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 0
[1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 1
[1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 3
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 2
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 3
[1,1,0,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 4
[1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 3
[1,1,1,0,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 4
[1,1,1,0,1,0,0,0]
=> [.,[.,[[.,.],.]]]
=> 5
[1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,.]]]]
=> 6
[1,0,1,0,1,0,1,0,1,0]
=> [[[[[.,.],.],.],.],.]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 5
[1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> 6
[1,1,0,1,1,1,0,0,0,0]
=> [.,[[.,.],[.,[.,.]]]]
=> 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[[.,.],.],.],.],[.,.]],.]
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],.],.],.],.]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[.,.]],.],.],.],.],.]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[[.,.],.]],.],.],.],.]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[.,.],.],.]],.],.],.]
=> ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[[.,.],.],.],.]],.],.]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[.,[[[[[.,.],.],.],.],.]],.]
=> ? = 5
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,.]]],.],.],.],.]
=> ? = 3
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,.]]]],.],.],.]
=> ? = 6
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> ? = 10
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 21
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> ? = 12
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> ? = 3
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [[.,.],[[[.,[.,.]],[.,.]],[.,.]]]
=> ? = 9
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> ? = 13
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]
=> ? = 13
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [[[.,[.,.]],[.,.]],[[.,.],[.,.]]]
=> ? = 6
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]
=> ? = 8
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> ? = 8
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 16
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[[[.,.],.],.],.]],.],.],.]
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 6
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> ? = 12
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [[[.,[[.,.],[.,.]]],[.,.]],[.,.]]
=> ? = 6
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> ? = 20
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> ? = 3
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]
=> ? = 9
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 13
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]
=> ? = 9
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]
=> ? = 12
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]
=> ? = 13
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> ? = 20
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> ? = 8
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]
=> ? = 16
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> ? = 10
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> ? = 13
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> ? = 13
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> ? = 15
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 16
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> ? = 18
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> ? = 20
Description
The sum of the sizes of the right subtrees of a binary tree. This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the 312-avoiding permutation corresponding to the binary tree. It is also the sum of all heights j of the coordinates (i,j) of the Dyck path corresponding to the binary tree.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000041The number of nestings of a perfect matching. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000057The Shynar inversion number of a standard tableau. St000081The number of edges of a graph. St000006The dinv of a Dyck path. St001397Number of pairs of incomparable elements in a finite poset. St000332The positive inversions of an alternating sign matrix. St001671Haglund's hag of a permutation. St000795The mad of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001428The number of B-inversions of a signed permutation. St000833The comajor index of a permutation. St000004The major index of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000005The bounce statistic of a Dyck path. St000042The number of crossings of a perfect matching. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. St000156The Denert index of a permutation. St000305The inverse major index of a permutation. St000796The stat' of a permutation. St000798The makl of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001311The cyclomatic number of a graph. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001718The number of non-empty open intervals in a poset. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000080The rank of the poset. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000450The number of edges minus the number of vertices plus 2 of a graph. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St000232The number of crossings of a set partition. St000359The number of occurrences of the pattern 23-1. St001866The nesting alignments of a signed permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St001433The flag major index of a signed permutation. St001822The number of alignments of a signed permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.