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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000659
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000836
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [1,2,4,6,3,5] => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [1,2,6,3,4,5] => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [1,3,4,6,2,5] => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [1,3,5,2,4,6] => 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [1,3,6,2,4,5] => 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => [1,4,6,2,3,5] => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => [1,5,2,3,4,6] => 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => [1,6,2,3,4,5] => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [2,3,4,6,1,5] => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [2,3,5,1,4,6] => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => [2,3,6,1,4,5] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [2,4,1,3,5,6] => 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => [2,4,6,1,3,5] => 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => [2,5,1,3,4,6] => 2 = 1 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => [2,6,1,3,4,5] => 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [3,4,6,1,2,5] => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [3,5,1,2,4,6] => 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => [3,6,1,2,4,5] => 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [4,1,2,3,5,6] => 1 = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [4,6,1,2,3,5] => 2 = 1 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => 1 = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [1,2,3,5,7,4,6] => 2 = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [1,2,3,7,4,5,6] => 1 = 0 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [1,2,4,5,7,3,6] => 2 = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [1,2,4,6,3,5,7] => 2 = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [1,2,4,7,3,5,6] => 2 = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [1,2,5,7,3,4,6] => 2 = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [1,2,6,3,4,5,7] => 1 = 0 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [1,2,7,3,4,5,6] => 1 = 0 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [1,3,4,5,7,2,6] => 2 = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [1,3,4,6,2,5,7] => 2 = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [1,3,4,7,2,5,6] => 2 = 1 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [1,3,5,2,4,6,7] => 2 = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [1,3,5,7,2,4,6] => 2 = 1 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [1,3,6,2,4,5,7] => 2 = 1 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [1,3,7,2,4,5,6] => 2 = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => [1,4,5,7,2,3,6] => 2 = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => [1,4,6,2,3,5,7] => 2 = 1 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [1,4,7,2,3,5,6] => ? = 1 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [1,5,2,3,4,6,7] => ? = 0 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [1,5,7,2,3,4,6] => ? = 1 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 0 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [1,7,2,3,4,5,6] => ? = 0 + 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => [2,3,4,5,7,1,6] => ? = 1 + 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => [2,3,4,6,1,5,7] => ? = 1 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => [2,3,4,7,1,5,6] => ? = 1 + 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => ? = 1 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => ? = 1 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => ? = 1 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,5,7,6] => [2,3,7,1,4,5,6] => ? = 1 + 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 1 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => ? = 1 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => ? = 1 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => ? = 1 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => ? = 1 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => ? = 1 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => ? = 1 + 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ? = 1 + 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ? = 1 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ? = 1 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ? = 2 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => ? = 1 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ? = 1 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 0 + 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 1 + 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 0 + 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1 + 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 0 + 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 0 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => [1,2,3,4,6,8,5,7] => ? = 1 + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [1,2,3,5,6,8,4,7] => ? = 1 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,6,7,5,8] => [1,2,3,5,7,4,6,8] => ? = 1 + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,8,7] => [1,2,3,5,8,4,6,7] => ? = 1 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,4,1,5,7,6,8] => [1,2,3,7,4,5,6,8] => ? = 0 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => [1,2,4,5,6,8,3,7] => ? = 1 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,5,6,7,4,8] => [1,2,4,5,7,3,6,8] => ? = 1 + 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ? => ? = 1 + 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 1 + 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6,8] => [1,2,4,7,3,5,6,8] => ? = 1 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,5,4,6,8,7] => ? => ? = 1 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,7,5,8] => [1,2,5,7,3,4,6,8] => ? = 1 + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,8,7] => [1,2,5,8,3,4,6,7] => ? = 1 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ? => ? = 0 + 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => [1,3,4,5,6,8,2,7] => ? = 1 + 1
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000495
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000495: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000495: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,1,3] => 3 = 1 + 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => 2 = 0 + 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => 3 = 1 + 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => 2 = 0 + 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 3 = 1 + 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 3 = 1 + 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => 3 = 1 + 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => 3 = 1 + 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 2 = 0 + 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 2 = 0 + 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [1,2,4,6,3,5] => 3 = 1 + 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [1,2,6,3,4,5] => 2 = 0 + 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [1,3,4,6,2,5] => 3 = 1 + 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [1,3,5,2,4,6] => 3 = 1 + 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [1,3,6,2,4,5] => 3 = 1 + 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => [1,4,6,2,3,5] => 3 = 1 + 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => [1,5,2,3,4,6] => 2 = 0 + 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => [1,6,2,3,4,5] => 2 = 0 + 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [2,3,4,6,1,5] => 3 = 1 + 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [2,3,5,1,4,6] => 3 = 1 + 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => [2,3,6,1,4,5] => 3 = 1 + 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => [2,4,1,3,5,6] => 3 = 1 + 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => [2,4,6,1,3,5] => 3 = 1 + 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => [2,5,1,3,4,6] => 3 = 1 + 2
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,4,6,5] => [2,6,1,3,4,5] => 3 = 1 + 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => [3,4,6,1,2,5] => 3 = 1 + 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => [3,5,1,2,4,6] => 3 = 1 + 2
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => [3,6,1,2,4,5] => 3 = 1 + 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => [4,1,2,3,5,6] => 2 = 0 + 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => [4,6,1,2,3,5] => 3 = 1 + 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => 2 = 0 + 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => 2 = 0 + 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,1,6,7,5] => [1,2,3,5,7,4,6] => 3 = 1 + 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,4,1,5,7,6] => [1,2,3,7,4,5,6] => 2 = 0 + 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,4] => [1,2,4,5,7,3,6] => 3 = 1 + 2
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,1,5,6,4,7] => [1,2,4,6,3,5,7] => 3 = 1 + 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,7,6] => [1,2,4,7,3,5,6] => 3 = 1 + 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,3,1,4,6,7,5] => [1,2,5,7,3,4,6] => 3 = 1 + 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => [1,2,6,3,4,5,7] => 2 = 0 + 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,3,1,4,5,7,6] => [1,2,7,3,4,5,6] => 2 = 0 + 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,3] => [1,3,4,5,7,2,6] => 3 = 1 + 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => [1,3,4,6,2,5,7] => 3 = 1 + 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => [1,3,4,7,2,5,6] => 3 = 1 + 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => [1,3,5,2,4,6,7] => 3 = 1 + 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => [1,3,5,7,2,4,6] => 3 = 1 + 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => [1,3,6,2,4,5,7] => 3 = 1 + 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => [1,3,7,2,4,5,6] => 3 = 1 + 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => [1,4,5,7,2,3,6] => 3 = 1 + 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => [1,4,6,2,3,5,7] => 3 = 1 + 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => [1,4,7,2,3,5,6] => ? = 1 + 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => [1,5,2,3,4,6,7] => ? = 0 + 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => [1,5,7,2,3,4,6] => ? = 1 + 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 0 + 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => [1,7,2,3,4,5,6] => ? = 0 + 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,7,2] => [2,3,4,5,7,1,6] => ? = 1 + 2
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => [2,3,4,6,1,5,7] => ? = 1 + 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,4,5,2,7,6] => [2,3,4,7,1,5,6] => ? = 1 + 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => ? = 1 + 2
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => ? = 1 + 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => ? = 1 + 2
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,3,4,2,5,7,6] => [2,3,7,1,4,5,6] => ? = 1 + 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 1 + 2
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => ? = 1 + 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => ? = 1 + 2
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => ? = 1 + 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => ? = 1 + 2
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => ? = 1 + 2
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => ? = 1 + 2
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => ? = 1 + 2
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ? = 1 + 2
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ? = 1 + 2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ? = 1 + 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => ? = 1 + 2
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ? = 2 + 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => ? = 1 + 2
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ? = 1 + 2
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 0 + 2
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 1 + 2
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 1 + 2
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ? = 1 + 2
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 0 + 2
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1 + 2
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 0 + 2
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 0 + 2
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,4,5,1,7,8,6] => [1,2,3,4,6,8,5,7] => ? = 1 + 2
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,6,7,8,5] => [1,2,3,5,6,8,4,7] => ? = 1 + 2
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,3,4,1,6,7,5,8] => [1,2,3,5,7,4,6,8] => ? = 1 + 2
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,8,7] => [1,2,3,5,8,4,6,7] => ? = 1 + 2
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,4,1,5,7,6,8] => [1,2,3,7,4,5,6,8] => ? = 0 + 2
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,5,6,7,8,4] => [1,2,4,5,6,8,3,7] => ? = 1 + 2
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,3,1,5,6,7,4,8] => [1,2,4,5,7,3,6,8] => ? = 1 + 2
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,6,4,8,7] => ? => ? = 1 + 2
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 1 + 2
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,7,6,8] => [1,2,4,7,3,5,6,8] => ? = 1 + 2
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,5,4,6,8,7] => ? => ? = 1 + 2
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,7,5,8] => [1,2,5,7,3,4,6,8] => ? = 1 + 2
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,8,7] => [1,2,5,8,3,4,6,7] => ? = 1 + 2
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,1,4,6,5,7,8] => ? => ? = 0 + 2
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,5,6,7,8,3] => [1,3,4,5,6,8,2,7] => ? = 1 + 2
Description
The number of inversions of distance at most 2 of a permutation.
An inversion of a permutation $\pi$ is a pair $i < j$ such that $\sigma(i) > \sigma(j)$. Let $j-i$ be the distance of such an inversion. Then inversions of distance at most 1 are then exactly the descents of $\pi$, see [[St000021]]. This statistic counts the number of inversions of distance at most 2.
Matching statistic: St000353
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => ? = 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ? = 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => ? = 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => ? = 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => ? = 0
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,6,3,5,4,2,1] => ? = 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => ? = 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => ? = 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ? = 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => ? = 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,7,3,4,6,5,1] => ? = 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? = 0
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => 0
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ? = 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ? = 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ? = 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ? = 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => ? = 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ? = 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ? = 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ? = 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? = 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ? = 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ? = 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? = 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => ? = 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ? = 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? = 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ? = 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => 0
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => 0
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => 0
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [8,2,7,6,5,4,3,1] => ? = 1
[1,1,1,2,1,1,1] => [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]
=> [8,3,7,6,5,4,2,1] => ? = 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,8,3,7,6,5,4,1] => ? = 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [8,2,3,7,6,5,4,1] => ? = 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,4,8,7,6,5,1] => ? = 0
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [8,7,3,6,5,4,2,1] => ? = 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,8,4,7,6,5,3,1] => ? = 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [8,2,4,7,6,5,3,1] => ? = 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [8,3,4,7,6,5,2,1] => ? = 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [8,2,3,4,7,6,5,1] => ? = 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ? = 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,8,4,7,6,5,1] => ? = 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [2,8,3,4,7,6,5,1] => ? = 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,8,7,6,1] => ? = 0
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000455
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 - 1
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0 - 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 - 1
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 - 1
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? = 0 - 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 1 - 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 0 - 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 0 - 1
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1 - 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1 - 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1 - 1
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 - 1
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1 - 1
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 - 1
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0 - 1
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],4)
=> ? = 0 - 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? = 1 - 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? = 0 - 1
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? = 0 - 1
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1 - 1
[1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1 - 1
[1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1 - 1
[1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 0 - 1
[1,4,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1 - 1
[1,4,2] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 0 - 1
[1,5,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 0 - 1
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,1,1] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[3,1,1,1,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[3,1,1,2] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[3,1,2,1] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[3,2,1,1] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 2 - 1
[3,2,2] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[3,3,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[3,4] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 0 - 1
[4,1,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[4,1,2] => [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[4,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[4,3] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 0 - 1
[5,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1 - 1
[5,2] => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 0 - 1
[6,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 0 - 1
[1,1,1,1,2,1,1] => [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,1,1,2,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,1,1,2,1,2] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,1,1,2,2,1] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,1,1,3,2] => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 0 - 1
[1,1,2,1,1,1,1] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,1,2,1,1,2] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
[1,1,2,1,2,1] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000092
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 2 = 1 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => 2 = 1 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1 = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => ? = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => ? = 0 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ? = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => ? = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => ? = 0 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => ? = 0 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,6,3,5,4,2,1] => ? = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => ? = 1 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => ? = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => ? = 1 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ? = 1 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => ? = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => ? = 1 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,7,3,4,6,5,1] => ? = 1 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? = 0 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => ? = 1 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => ? = 0 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => ? = 0 + 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ? = 1 + 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 1 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ? = 1 + 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? = 1 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ? = 1 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ? = 1 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => ? = 1 + 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => ? = 1 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ? = 1 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ? = 1 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ? = 1 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ? = 1 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ? = 1 + 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? = 1 + 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 1 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => ? = 1 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 2 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? = 1 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ? = 1 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 0 + 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ? = 1 + 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 0 + 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ? = 1 + 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000243
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 67%
Values
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,6,4,5,2] => 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 2 = 1 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => 2 = 1 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 1 = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => 2 = 1 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1 = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => ? = 1 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => ? = 0 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => ? = 1 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,7,3,6,5,4,2] => ? = 1 + 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,7,6,5,1] => ? = 0 + 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => ? = 0 + 1
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [7,6,3,5,4,2,1] => ? = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => ? = 1 + 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => ? = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => ? = 1 + 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ? = 1 + 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 1 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => ? = 1 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => ? = 1 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,7,3,4,6,5,1] => ? = 1 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ? = 0 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => ? = 1 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => ? = 0 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => ? = 0 + 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [7,6,4,5,3,2,1] => ? = 1 + 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 1 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ? = 1 + 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? = 1 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ? = 1 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ? = 1 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => ? = 1 + 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => ? = 1 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ? = 1 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ? = 1 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? = 1 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ? = 1 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ? = 1 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => ? = 1 + 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? = 1 + 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 1 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => ? = 1 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 2 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? = 1 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ? = 1 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ? = 0 + 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => ? = 1 + 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => ? = 1 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => ? = 1 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ? = 0 + 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ? = 1 + 1
Description
The number of cyclic valleys and cyclic peaks of a permutation.
This is given by the number of indices $i$ such that $\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices $i$ such that $\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.
Matching statistic: St001691
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
Description
The number of kings in a graph.
A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.
Matching statistic: St000456
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00274: Graphs —block-cut tree⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 0
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> ? = 0
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ? = 1
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 0
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> ? = 0
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 1
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 1
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],1)
=> ? = 0
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],4)
=> ? = 0
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> ? = 1
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? = 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? = 0
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> ? = 0
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 1
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ? = 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ? = 1
[1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ? = 1
[1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1
[1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1
[1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1
[1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 0
[1,4,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 1
[1,4,2] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 0
[1,5,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],2)
=> ? = 0
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[2,3,1,1] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(1,2)],3)
=> 1
[3,1,1,1,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1
[3,1,1,2] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1
[3,1,2,1] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1
[3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1
[3,2,1,1] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 2
[3,2,2] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1
[3,3,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ? = 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001305
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 0
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 0
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 0
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
Description
The number of induced cycles on four vertices in a graph.
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