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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000668
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 2
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 3
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [3,3,1]
=> [3,1]
=> 3
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> [2,2,2]
=> 2
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [3,2,2]
=> [2,2]
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> [2]
=> 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [3,3,2]
=> [3,2]
=> 6
[3,2,2] => [[5,4,3],[3,2]]
=> [3,2]
=> [2]
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> [4,2]
=> [2]
=> 2
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> [3,3]
=> 3
[4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> [3]
=> 3
[4,2,1] => [[5,5,4],[4,3]]
=> [4,3]
=> [3]
=> 3
[5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> [4]
=> 4
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000120
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 27%●distinct values known / distinct values provided: 25%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 27%●distinct values known / distinct values provided: 25%
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
=> [4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 1
[3,1,3] => [[5,3,3],[2,2]]
=> [5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 + 1
[3,2,2] => [[5,4,3],[3,2]]
=> [5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[3,3,1] => [[5,5,3],[4,2]]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[4,1,2] => [[5,4,4],[3,3]]
=> [5,4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 3 + 1
[4,2,1] => [[5,5,4],[4,3]]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]]
=> [3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]]
=> [3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]]
=> [4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
=> [3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]]
=> [3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 1
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]]
=> [3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]]
=> [4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]]
=> [4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]]
=> [3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2 + 1
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]]
=> [4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]]
=> [4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]]
=> [4,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]]
=> [4,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 2 + 1
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]]
=> [4,4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,3,1,3] => [[5,3,3,1],[2,2]]
=> [5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]]
=> [4,4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 + 1
[1,3,2,2] => [[5,4,3,1],[3,2]]
=> [5,4,3,1]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,3,1] => [[5,5,3,1],[4,2]]
=> [5,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 2 + 1
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
=> [4,4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,4,1,2] => [[5,4,4,1],[3,3]]
=> [5,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 3 + 1
[1,4,2,1] => [[5,5,4,1],[4,3]]
=> [5,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]]
=> [3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]]
=> [3,3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 1
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]]
=> [3,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]]
=> [4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 1
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]]
=> [4,4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
=> [3,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]]
=> [4,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]]
=> [5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]]
=> [4,4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[2,1,3,2] => [[5,4,2,2],[3,1,1]]
=> [5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,1,3] => [[5,3,3,2],[2,2,1]]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,2,2] => [[5,4,3,2],[3,2,1]]
=> [5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000451
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 38%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 38%
Values
[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] => 3 = 1 + 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 4 = 2 + 2
[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] => 3 = 1 + 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => 4 = 2 + 2
[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] => 3 = 1 + 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 3 = 1 + 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 3 = 1 + 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => 4 = 2 + 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 4 = 2 + 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => 4 = 2 + 2
[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] => 4 = 2 + 2
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 5 = 3 + 2
[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 + 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => ? = 2 + 2
[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 + 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => 3 = 1 + 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 1 + 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => ? = 2 + 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => ? = 2 + 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 2 + 2
[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] => ? = 2 + 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ? = 3 + 2
[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
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => 3 = 1 + 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 1 + 2
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => 3 = 1 + 2
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ? = 2 + 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => 3 = 1 + 2
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 1 + 2
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ? = 2 + 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => 4 = 2 + 2
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => ? = 2 + 2
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ? = 3 + 2
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ? = 2 + 2
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => 4 = 2 + 2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 2 + 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => 4 = 2 + 2
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ? = 6 + 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => 4 = 2 + 2
[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] => ? = 2 + 2
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ? = 3 + 2
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? = 3 + 2
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ? = 3 + 2
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? = 4 + 2
[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 + 2
[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 + 2
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => 3 = 1 + 2
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [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 + 2
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [8,2,4,7,6,5,3,1] => ? = 2 + 2
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ? = 2 + 2
[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] => ? = 2 + 2
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,7,4,6,5,3,2,1] => ? = 1 + 2
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => 3 = 1 + 2
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,8,7,4,6,5,3,1] => ? = 1 + 2
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,2,7,4,6,5,3,1] => 4 = 2 + 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,8,5,7,6,4,2] => ? = 1 + 2
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,8,5,7,6,4,1] => ? = 1 + 2
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [8,3,7,4,6,5,2,1] => 4 = 2 + 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,8,3,5,7,6,4,2] => ? = 2 + 2
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,8,3,5,7,6,4,1] => ? = 2 + 2
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [8,2,3,5,7,6,4,1] => ? = 3 + 2
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,8,4,5,7,6,3,2] => ? = 2 + 2
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,8,4,5,7,6,3,1] => ? = 2 + 2
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ? = 2 + 2
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [8,2,4,5,7,6,3,1] => ? = 6 + 2
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,8,4,5,7,6,2] => ? = 2 + 2
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,8,4,5,7,6,1] => ? = 2 + 2
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [8,3,4,5,7,6,2,1] => ? = 3 + 2
[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,8,3,4,5,7,6,2] => ? = 3 + 2
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [2,8,3,4,5,7,6,1] => ? = 3 + 2
[2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8,7,6,4,5,3,2,1] => ? = 1 + 2
[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,8,7,5,6,4,3,2] => 3 = 1 + 2
[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,7,5,6,4,3,1] => ? = 1 + 2
[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [8,2,7,5,6,4,3,1] => ? = 2 + 2
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,3,8,7,5,6,4,2] => ? = 1 + 2
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,8,5,6,7,1] => 4 = 2 + 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001638
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 38%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001638: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 38%
Values
[2,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[3,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1,1] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,3,1,1] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[2,1,1,1,1] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[2,1,1,2] => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 1
[2,1,2,1] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1,1] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,1,1,1] => [1,1,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[3,1,2] => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[3,2,1] => [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[4,1,1] => [1,1,1,3] => [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)
=> ? = 3
[1,1,2,1,1,1] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[1,1,3,1,1] => [3,1,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,2,1,1,1,1] => [2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[1,2,1,1,2] => [2,4,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[1,2,1,2,1] => [2,3,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[1,2,2,1,1] => [2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,3,1,1,1] => [2,1,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,3,1,2] => [2,1,3,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,3,2,1] => [2,1,2,2] => [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
[1,4,1,1] => [2,1,1,3] => [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)
=> ? = 3
[2,1,1,1,1,1] => [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[2,1,1,1,2] => [1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 1
[2,1,1,2,1] => [1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,1,1,3] => [1,4,1,1] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,2,1,1] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,1,2,2] => [1,3,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,3,1] => [1,3,1,2] => [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
[2,2,1,1,1] => [1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[2,2,1,2] => [1,2,3,1] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[2,2,2,1] => [1,2,2,2] => [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)
=> ? = 2
[2,3,1,1] => [1,2,1,3] => [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)
=> ? = 3
[3,1,1,1,1] => [1,1,5] => [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)
=> ? = 2
[3,1,1,2] => [1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,1,2,1] => [1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[3,1,3] => [1,1,3,1,1] => [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)
=> ? = 2
[3,2,1,1] => [1,1,2,3] => [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)
=> ? = 6
[3,2,2] => [1,1,2,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[3,3,1] => [1,1,2,1,2] => [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)
=> ? = 2
[4,1,1,1] => [1,1,1,4] => [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)
=> ? = 3
[4,1,2] => [1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,2,1] => [1,1,1,2,2] => [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)
=> ? = 3
[5,1,1] => [1,1,1,1,3] => [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)
=> ? = 4
[1,1,1,2,1,1,1] => [4,4] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,1,1,1,1] => [3,5] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,1,1,2] => [3,4,1] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,1,2,1] => [3,3,2] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,2,1,1] => [3,2,3] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,3,1,2] => [3,1,3,1] => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,3,2,1] => [3,1,2,2] => [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,1,1,1,1,1] => [2,6] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,2,1,1,1,2] => [2,5,1] => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,1,1,2,1] => [2,4,2] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,1,2,1,1] => [2,3,3] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,1,2,2] => [2,3,2,1] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,1,3,1] => [2,3,1,2] => [2,2,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,2,2,1,1,1] => [2,2,4] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,2,1,2] => [2,2,3,1] => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,2,2,1] => [2,2,2,2] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,2,3,1,1] => [2,2,1,3] => [3,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,3,1,1,2] => [2,1,4,1] => [1,2,1,4] => ([(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,3,1,2,1] => [2,1,3,2] => [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,3,1,3] => [2,1,3,1,1] => [1,2,1,3,1] => ([(0,7),(1,7),(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)
=> ? = 2
[1,3,2,1,1] => [2,1,2,3] => [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 6
[1,3,2,2] => [2,1,2,2,1] => [1,2,1,2,2] => ([(1,7),(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)
=> ? = 2
[1,3,3,1] => [2,1,2,1,2] => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(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)
=> ? = 2
[1,4,1,1,1] => [2,1,1,4] => [4,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[1,4,1,2] => [2,1,1,3,1] => [1,2,1,1,3] => ([(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)
=> ? = 3
[1,4,2,1] => [2,1,1,2,2] => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(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)
=> ? = 3
[2,1,1,1,1,1,1] => [1,7] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[2,1,1,1,1,2] => [1,6,1] => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,1,1,1,2,1] => [1,5,2] => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
Description
The book thickness of a graph.
The book thickness (or pagenumber, or stacknumber) of a graph is the minimal number of pages required for a book embedding of a graph.
Matching statistic: St000299
Values
[2,1,1,1] => ([(0,2),(0,3),(0,4),(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),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 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)
=> ([(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)
=> 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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[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)
=> ([(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)
=> 3 = 1 + 2
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[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),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[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)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[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)
=> ([(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)
=> 4 = 2 + 2
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[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),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[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)
=> ([(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
[1,1,3,1,1] => ([(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)
=> ([(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)
=> ? = 2 + 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)
=> ([(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
[1,2,1,1,2] => ([(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)
=> ([(1,4),(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
[1,2,1,2,1] => ([(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)
=> ([(0,3),(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)
=> 3 = 1 + 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)
=> ([(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)
=> ? = 2 + 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)
=> ([(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)
=> ? = 2 + 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)
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 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)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 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)
=> ([(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)
=> ? = 3 + 2
[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)
=> ([(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
[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)
=> ([(1,3),(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
[2,1,1,2,1] => ([(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),(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)
=> 3 = 1 + 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[2,1,2,1,1] => ([(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)
=> ([(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)
=> ? = 2 + 2
[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),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[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)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[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)
=> ([(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)
=> ? = 2 + 2
[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)
=> ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[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)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[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)
=> ([(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)
=> ? = 3 + 2
[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)
=> ([(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)
=> ? = 2 + 2
[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)
=> ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[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)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 2
[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)
=> ([(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)
=> ? = 6 + 2
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[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)
=> ([(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)
=> ? = 3 + 2
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[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)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 + 2
[1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(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 + 2
[1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(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 + 2
[1,1,2,1,1,2] => ([(1,5),(1,6),(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 + 2
[1,1,2,1,2,1] => ([(0,7),(1,5),(1,6),(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 + 2
[1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(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)
=> ?
=> ? = 2 + 2
[1,1,3,1,2] => ([(1,6),(1,7),(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)
=> ?
=> ? = 2 + 2
[1,1,3,2,1] => ([(0,7),(1,6),(1,7),(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)
=> ?
=> ? = 2 + 2
[1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(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 + 2
[1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(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 + 2
[1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(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 + 2
[1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(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)
=> ?
=> ? = 2 + 2
[1,2,1,2,2] => ([(1,7),(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 + 2
[1,2,1,3,1] => ([(0,7),(1,7),(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 + 2
[1,2,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(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)
=> ?
=> ? = 2 + 2
[1,2,2,1,2] => ([(1,6),(1,7),(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)
=> ?
=> ? = 2 + 2
[1,2,2,2,1] => ([(0,7),(1,6),(1,7),(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)
=> ?
=> ? = 2 + 2
[1,2,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(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)
=> ?
=> ? = 3 + 2
[1,3,1,1,2] => ([(1,5),(1,6),(1,7),(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)
=> ?
=> ? = 2 + 2
[1,3,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(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)
=> ?
=> ? = 2 + 2
[1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 2
[1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(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)
=> ?
=> ? = 6 + 2
[1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 2
[1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 2
[1,4,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(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)
=> ?
=> ? = 3 + 2
[1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3 + 2
[1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3 + 2
[2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(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 + 2
[2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(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 + 2
Description
The number of nonisomorphic vertex-induced subtrees.
Matching statistic: St000710
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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
St000710: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Values
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 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
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [1,3,4,6,2,5] => 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
[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] => 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] => 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] => 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
[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
[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
[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
[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
[1,1,2,1,1,1] => [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] => 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
[1,2,1,1,1,1] => [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] => 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] => 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] => 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
[1,3,1,1,1] => [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
[1,3,1,2] => [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
[1,3,2,1] => [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] => ? = 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] => ? = 3
[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
[2,1,1,1,2] => [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
[2,1,1,2,1] => [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
[2,1,1,3] => [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
[2,1,2,1,1] => [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] => ? = 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
[2,1,3,1] => [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
[2,2,1,1,1] => [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] => ? = 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] => ? = 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] => ? = 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] => ? = 3
[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] => ? = 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] => ? = 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] => ? = 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] => ? = 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] => ? = 6
[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] => ? = 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] => ? = 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] => ? = 3
[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] => ? = 3
[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] => ? = 3
[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] => ? = 4
[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
[1,1,2,1,1,1,1] => [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
[1,1,2,1,1,2] => [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
[1,1,2,1,2,1] => [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
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,7,8,6] => ? => ? = 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] => ? = 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] => ? = 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
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,7,3,8] => [1,3,4,5,7,2,6,8] => ? = 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,6,3,8,7] => [1,3,4,5,8,2,6,7] => ? = 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3,7,8,6] => ? => ? = 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6,8] => ? => ? = 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,3,6,8,7] => [1,3,4,8,2,5,6,7] => ? = 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [1,3,5,6,8,2,4,7] => ? = 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,6,7,5,8] => [1,3,5,7,2,4,6,8] => ? = 2
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [1,3,5,8,2,4,6,7] => ? = 2
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [1,3,6,8,2,4,5,7] => ? = 3
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,5,6,7,4,8] => ? => ? = 2
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,6,4,8,7] => [1,4,5,8,2,3,6,7] => ? = 2
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,5,6,4,7,8] => ? => ? = 2
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,7,8,6] => [1,4,6,8,2,3,5,7] => ? = 6
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6,8] => ? => ? = 2
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4,6,8,7] => [1,4,8,2,3,5,6,7] => ? = 2
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,1,3,4,6,7,8,5] => [1,5,6,8,2,3,4,7] => ? = 3
[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,1,3,4,6,7,5,8] => [1,5,7,2,3,4,6,8] => ? = 3
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,8,7] => [1,5,8,2,3,4,6,7] => ? = 3
Description
The number of big deficiencies of a permutation.
A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$.
This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St000711
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 38%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 38%
Values
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => 3 = 2 + 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]
=> [4,6,1,2,3,5] => 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => 3 = 2 + 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]
=> [5,6,1,2,3,4] => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => 3 = 2 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => 3 = 2 + 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]
=> [2,4,5,6,1,3] => 3 = 2 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => 4 = 3 + 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]
=> [4,7,1,2,3,5,6] => ? = 1 + 1
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,7,1,2,5,6] => ? = 2 + 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]
=> [5,7,1,2,3,4,6] => ? = 1 + 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,2,3,4,6] => ? = 1 + 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,5,7,1,3,4,6] => ? = 1 + 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,5,7,1,2,4,6] => ? = 2 + 1
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,7,1,2,3,6] => ? = 2 + 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,7,2,3,6] => 3 = 2 + 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]
=> [2,4,5,7,1,3,6] => ? = 2 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ? = 3 + 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]
=> [6,7,1,2,3,4,5] => ? = 1 + 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => ? = 1 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,6,7,1,3,4,5] => ? = 1 + 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => 2 = 1 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,6,7,1,2,4,5] => ? = 2 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,4,5] => 2 = 1 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,6,7,1,4,5] => ? = 1 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,6,7,1,2,3,5] => ? = 2 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,2,3,5] => ? = 2 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ? = 2 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ? = 3 + 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 2 + 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,2,3,4] => ? = 2 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => ? = 2 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => 3 = 2 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,1,2,4] => ? = 6 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => 3 = 2 + 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]
=> [2,3,5,6,7,1,4] => ? = 2 + 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ? = 3 + 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,5,6,7,2,3] => 4 = 3 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,4,5,6,7,1,3] => ? = 3 + 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,1,2] => ? = 4 + 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]
=> [4,8,1,2,3,5,6,7] => ? = 1 + 1
[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]
=> [5,8,1,2,3,4,6,7] => ? = 1 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,2,3,4,6,7] => ? = 1 + 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ? = 1 + 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,5,8,1,2,4,6,7] => ? = 2 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ? = 2 + 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]
=> [2,4,5,8,1,3,6,7] => ? = 2 + 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => ? = 1 + 1
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => ? = 1 + 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,6,8,1,3,4,5,7] => ? = 1 + 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,6,8,1,2,4,5,7] => ? = 2 + 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,6,8,2,4,5,7] => ? = 1 + 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,6,8,1,4,5,7] => ? = 1 + 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [4,6,8,1,2,3,5,7] => ? = 2 + 1
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,4,6,8,2,3,5,7] => ? = 2 + 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => ? = 2 + 1
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,6,8,1,2,5,7] => ? = 3 + 1
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,5,6,8,2,3,4,7] => ? = 2 + 1
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,5,6,8,1,3,4,7] => ? = 2 + 1
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,5,6,8,3,4,7] => ? = 2 + 1
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,5,6,8,1,2,4,7] => ? = 6 + 1
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,5,6,8,2,4,7] => ? = 2 + 1
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,5,6,8,1,4,7] => ? = 2 + 1
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,6,8,1,2,3,7] => ? = 3 + 1
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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),(1,2),(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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> 1
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> 1
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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),(2,3),(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)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[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)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[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)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[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)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[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)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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),(1,2),(1,3),(1,4),(2,3),(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),(1,2),(1,3),(2,3)],4)
=> ? = 2
[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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2
[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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[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)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[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)
=> ([(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)
=> ? = 2
[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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6
[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)
=> ([(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)
=> ? = 2
[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)
=> ([(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)
=> ? = 2
[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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[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)
=> ([(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)
=> ? = 3
[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)
=> ([(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)
=> ? = 3
[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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,2,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,1,1,1,1] => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,1,1,2] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,1,2,1] => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,2,1,1] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,1,3,1,2] => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,1,3,2,1] => [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,2,1,1,1,1,1] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,2,1,1,1,2] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,2,1,1,2,1] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,2,1,2,1,1] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,2,1,2,2] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,2,1,3,1] => [2,1,3,2] => ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,2,2,1,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,2,2,1,2] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,2,2,2,1] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,2,3,1,1] => [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
[1,3,1,1,2] => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,3,1,2,1] => [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,3,1,3] => [1,1,3,1,2] => ([(1,6),(1,7),(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)
=> ?
=> ? = 2
[1,3,2,1,1] => [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
[1,3,2,2] => [1,2,2,1,2] => ([(1,6),(1,7),(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)
=> ?
=> ? = 2
[5,1,1,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001330
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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),(1,2),(1,3),(2,3)],4)
=> ? = 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> 2 = 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)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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),(2,3),(2,4),(3,4)],5)
=> ? = 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)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 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)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 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)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 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)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 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)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 2 = 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 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)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 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)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 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)
=> ([(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)
=> ? = 2 + 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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 + 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)
=> ([(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)
=> ? = 2 + 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)
=> ([(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)
=> ? = 2 + 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 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)
=> ([(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)
=> ? = 3 + 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)
=> ([(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)
=> ? = 3 + 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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,1,2,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 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
[1,1,2,2,1,1] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,1,3,1,2] => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,1,3,2,1] => [2,2,1,3] => ([(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,2,1,1,1,1,1] => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 1
[1,2,1,1,1,2] => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 1
[1,2,1,1,2,1] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 1
[1,2,1,2,1,1] => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,2,1,2,2] => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 1
[1,2,1,3,1] => [2,1,3,2] => ([(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 1
[1,2,2,1,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,2,2,1,2] => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,2,2,2,1] => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,2,3,1,1] => [3,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3 + 1
[1,3,1,1,2] => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,3,1,2,1] => [2,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 1
[1,3,1,3] => [1,1,3,1,2] => ([(1,6),(1,7),(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)
=> ?
=> ? = 2 + 1
[1,3,2,1,1] => [3,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6 + 1
[1,3,2,2] => [1,2,2,1,2] => ([(1,6),(1,7),(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)
=> ?
=> ? = 2 + 1
[5,1,1,1] => [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001744
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 38%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 38%
Values
[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
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 2
[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
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => 2
[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]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,6,3,4,5,2] => 2
[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
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => 3
[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
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => ? = 2
[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]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,7,4,6,5,3,2] => ? = 1
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 1
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => ? = 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => ? = 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,7,3,4,6,5,2] => ? = 2
[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] => ? = 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ? = 3
[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]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? = 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => ? = 2
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ? = 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => ? = 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,7,3,5,6,4,2] => ? = 2
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => ? = 2
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ? = 3
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [7,6,3,4,5,2,1] => ? = 2
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,7,4,5,6,3,2] => ? = 2
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,7,4,5,6,3,1] => ? = 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => 2
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ? = 6
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => 2
[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] => ? = 2
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,3,4,5,6,2,1] => ? = 3
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,7,3,4,5,6,2] => ? = 3
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ? = 3
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ? = 4
[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
[1,1,2,1,1,1,1] => [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]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => ? = 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,8,4,7,6,5,3,1] => ? = 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [8,2,4,7,6,5,3,1] => ? = 2
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,8,3,4,7,6,5,2] => ? = 2
[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] => ? = 2
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8,7,4,6,5,3,2,1] => ? = 1
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,8,7,4,6,5,3,2] => ? = 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,8,7,4,6,5,3,1] => ? = 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,2,7,4,6,5,3,1] => ? = 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,3,8,5,7,6,4,2] => ? = 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,8,5,7,6,4,1] => ? = 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [8,3,7,4,6,5,2,1] => ? = 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,8,3,5,7,6,4,2] => ? = 2
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,8,3,5,7,6,4,1] => ? = 2
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,1,0,0,0,0,0]
=> [8,2,3,5,7,6,4,1] => ? = 3
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,8,4,5,7,6,3,2] => ? = 2
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [2,8,4,5,7,6,3,1] => ? = 2
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,8,4,5,7,6,3] => ? = 2
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [8,2,4,5,7,6,3,1] => ? = 6
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,3,8,4,5,7,6,2] => ? = 2
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000028The number of stack-sorts needed to sort a permutation. St000359The number of occurrences of the pattern 23-1. St000872The number of very big descents of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000141The maximum drop size of a permutation. St000335The difference of lower and upper interactions. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001962The proper pathwidth of a graph. St000054The first entry of the permutation. St000443The number of long tunnels of a Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra.
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