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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
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(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,2] => [3,1] => [1,3] => [1,1,2] => 1
[1,1,2,1] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,1,1,2] => [4,1] => [1,4] => [1,1,1,2] => 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,3] => [3,1] => [1,3] => [1,1,2] => 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => [2,2,1] => 2
[1,1,3,1] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,1,1,1,2] => [5,1] => [1,5] => [1,1,1,1,2] => 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => [2,1,1,2] => 2
[1,1,1,1,3] => [4,1] => [1,4] => [1,1,1,2] => 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => [2,1,2,1] => 2
[1,1,1,2,2] => [3,2] => [2,3] => [1,1,2,1] => 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,4] => [3,1] => [1,3] => [1,1,2] => 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => [2,2,1,1] => 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [1,2,2] => 1
[1,1,2,3] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => [2,2,1] => 2
[1,1,3,2] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,4,1] => [2,1,1] => [1,2,1] => [2,2] => 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => [1,1,3] => 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => [2,3] => 2
[2,2,1,2] => [2,1,1] => [1,2,1] => [2,2] => 2
[2,2,2,1] => [3,1] => [1,3] => [1,1,2] => 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => [1,1,1,1,1,2] => 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => [2,1,1,1,2] => 2
[1,1,1,1,1,3] => [5,1] => [1,5] => [1,1,1,1,2] => 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => [2,1,1,2,1] => 2
[1,1,1,1,2,2] => [4,2] => [2,4] => [1,1,1,2,1] => 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => [2,1,1,2] => 2
[1,1,1,1,4] => [4,1] => [1,4] => [1,1,1,2] => 1
[1,1,1,2,1,1,1] => [3,1,3] => [3,3,1] => [2,1,2,1,1] => 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,3,1,1] => [3,1,2] => 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => [1,2,1,2] => 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => [2,1,2,1] => 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,5] => [3,1] => [1,3] => [1,1,2] => 1
[1,1,2,1,1,1,1] => [2,1,4] => [4,2,1] => [2,2,1,1,1] => 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => [1,3,2] => 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,2,1,1,1] => [4,2] => 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => [1,2,2,1] => 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
[1,1,2,4] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => [2,2,1,1] => 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => [3,2] => 3
Description
The first part of an integer composition.
Matching statistic: St000993
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000678
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 98%●distinct values known / distinct values provided: 83%
Values
[1,1,1,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,1,2] => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,2,2] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,4] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,4,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[2,2,1,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,2,2,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,3] => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,2,2] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,5] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,4] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,1,1,2,1,1,1] => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,3,1,1,1] => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,2,2,1,1,1] => [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[2,1,1,1,2,1,1,1] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,2,2,2] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,2,1,1,2,2,2] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,2,2,2,1,1,1,1] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,1,1,1,2,2,2] => [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,2,1,1,1,2,2,2] => [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,2,1,2,2,2] => [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,2,1,2,1,2,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[2,2,2,1,2,1,1,1] => [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[2,2,1,2,2,1,1,1] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,1,2,2,2,1,1,1] => [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,2,2,2,2,1,1,1] => [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,3,1] => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,5,1] => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,3,1,2,1,2,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,2,1,2,1,3,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,2,1,3,1,2,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,1,1,1,3] => [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,2,2,2,1,1,1] => [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000383
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 96%●distinct values known / distinct values provided: 83%
Values
[1,1,1,2] => [3,1] => [1,3] => [2,1,1] => 1
[1,1,2,1] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,1,1,2] => [4,1] => [1,4] => [2,1,1,1] => 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,3] => [3,1] => [1,3] => [2,1,1] => 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => [1,2,2] => 2
[1,1,3,1] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,1,1,1,2] => [5,1] => [1,5] => [2,1,1,1,1] => 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => [2,1,1,2] => 2
[1,1,1,1,3] => [4,1] => [1,4] => [2,1,1,1] => 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => [1,2,1,2] => 2
[1,1,1,2,2] => [3,2] => [2,3] => [1,2,1,1] => 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,4] => [3,1] => [1,3] => [2,1,1] => 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => [1,1,2,2] => 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [2,2,1] => 1
[1,1,2,3] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => [1,2,2] => 2
[1,1,3,2] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,4,1] => [2,1,1] => [1,2,1] => [2,2] => 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => [3,1,1] => 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => [3,2] => 2
[2,2,1,2] => [2,1,1] => [1,2,1] => [2,2] => 2
[2,2,2,1] => [3,1] => [1,3] => [2,1,1] => 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => [2,1,1,1,1,1] => 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => [2,1,1,1,2] => 2
[1,1,1,1,1,3] => [5,1] => [1,5] => [2,1,1,1,1] => 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => [1,2,1,1,2] => 2
[1,1,1,1,2,2] => [4,2] => [2,4] => [1,2,1,1,1] => 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => [2,1,1,2] => 2
[1,1,1,1,4] => [4,1] => [1,4] => [2,1,1,1] => 1
[1,1,1,2,1,1,1] => [3,1,3] => [3,3,1] => [1,1,2,1,2] => 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,3,1,1] => [2,1,3] => 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => [2,1,2,1] => 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => [1,2,1,2] => 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => [2,1,2] => 2
[1,1,1,5] => [3,1] => [1,3] => [2,1,1] => 1
[1,1,2,1,1,1,1] => [2,1,4] => [4,2,1] => [1,1,1,2,2] => 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => [2,3,1] => 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,2,1,1,1] => [2,4] => 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => [1,2,2,1] => 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
[1,1,2,4] => [2,1,1] => [1,2,1] => [2,2] => 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => [1,1,2,2] => 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => [2,3] => 3
[1,1,1,1,1,1,1,2] => [7,1] => [1,7] => [2,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,8] => [2,1,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,7,1] => [2,1,1,1,1,1,2] => ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => [1,7] => [2,1,1,1,1,1,1] => ? = 1
[1,1,1,1,1,2,1,2] => [5,1,1,1] => [1,5,1,1] => [2,1,1,1,3] => ? = 3
[1,1,1,1,2,1,2,1] => [4,1,1,1,1] => [1,4,1,1,1] => [2,1,1,4] => ? = 4
[1,1,1,1,2,1,2,3] => [4,1,1,1,1] => [1,4,1,1,1] => [2,1,1,4] => ? = 4
[1,1,1,1,3,2,1,2] => [4,1,1,1,1] => [1,4,1,1,1] => [2,1,1,4] => ? = 4
[1,1,2,1,2,3,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,2,5] => ? = 5
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => [1,2,1,1,1,2] => [2,5,1] => ? = 1
[1,1,3,2,1,2,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,2,5] => ? = 5
[1,1,3,2,1,1,1,2] => [2,1,1,3,1] => [1,2,1,1,3] => [2,4,1,1] => ? = 1
[2,1,2,2,1,1,1,2] => [1,1,2,3,1] => [1,1,1,2,3] => [4,2,1,1] => ? = 1
[2,1,2,1,1,2,1,2] => [1,1,1,2,1,1,1] => [1,1,1,1,2,1,1] => [5,3] => ? = 3
[3,1,1,2,1,2,1,1] => [1,2,1,1,1,2] => [2,1,2,1,1,1] => [1,3,4] => ? = 4
[1,1,1,1,1,1,1,1,1,2] => [9,1] => [1,9] => [2,1,1,1,1,1,1,1,1] => ? = 1
[2,2,1,1,2,1,2,1] => [2,2,1,1,1,1] => [1,2,2,1,1,1] => [2,2,4] => ? = 4
[2,1,2,2,1,1,2,1] => [1,1,2,2,1,1] => [1,1,1,2,2,1] => [4,2,2] => ? = 2
[2,1,1,2,1,2,1,2] => [1,2,1,1,1,1,1] => [1,1,2,1,1,1,1] => [3,5] => ? = 5
[2,1,1,1,2,2,2,1] => [1,3,3,1] => [1,1,3,3] => [3,1,2,1,1] => ? = 1
[1,2,1,2,2,1,2,1] => [1,1,1,2,1,1,1] => [1,1,1,1,2,1,1] => [5,3] => ? = 3
[1,2,1,1,2,2,1,2] => [1,1,2,2,1,1] => [1,1,1,2,2,1] => [4,2,2] => ? = 2
[1,2,1,1,2,2,2,1] => [1,1,2,3,1] => [1,1,1,2,3] => [4,2,1,1] => ? = 1
[1,2,1,1,1,2,2,2] => [1,1,3,3] => [3,1,1,3] => [1,1,4,1,1] => ? = 1
[1,2,2,1,2,1,2,1] => [1,2,1,1,1,1,1] => [1,1,2,1,1,1,1] => [3,5] => ? = 5
[1,1,2,2,1,2,1,2] => [2,2,1,1,1,1] => [1,2,2,1,1,1] => [2,2,4] => ? = 4
[1,1,2,1,2,1,2,2] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,2,5] => ? = 5
[1,1,2,1,2,2,2,1] => [2,1,1,3,1] => [1,2,1,1,3] => [2,4,1,1] => ? = 1
[1,1,1,2,1,2,2,2] => [3,1,1,3] => [3,3,1,1] => [1,1,2,1,3] => ? = 3
[1,1,1,2,1,2,1,2,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [2,1,6] => ? = 6
[2,1,1,1,1,2,2,1] => [1,4,2,1] => [1,1,4,2] => [3,1,1,2,1] => ? = 1
[2,2,1,2,1,2,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,2,5] => ? = 5
[2,2,2,1,2,1,1,1] => [3,1,1,3] => [3,3,1,1] => [1,1,2,1,3] => ? = 3
[2,1,2,2,2,1,1,1] => [1,1,3,3] => [3,1,1,3] => [1,1,4,1,1] => ? = 1
[1,1,1,1,1,1,1,3,1] => [7,1,1] => [1,7,1] => [2,1,1,1,1,1,2] => ? = 2
[1,1,1,1,1,1,1,5,1] => [7,1,1] => [1,7,1] => [2,1,1,1,1,1,2] => ? = 2
[1,1,1,3,1,2,1,2,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [2,1,6] => ? = 6
[1,1,1,2,1,2,1,3,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [2,1,6] => ? = 6
[1,1,1,2,1,3,1,2,1] => [3,1,1,1,1,1,1] => [1,3,1,1,1,1,1] => [2,1,6] => ? = 6
[1,1,1,1,1,1,1,1,3] => [8,1] => [1,8] => [2,1,1,1,1,1,1,1] => ? = 1
[1,2,2,2,1,1,1,2] => [1,3,3,1] => [1,1,3,3] => [3,1,2,1,1] => ? = 1
[2,2,1,2,1,1,1,2] => [2,1,1,3,1] => [1,2,1,1,3] => [2,4,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000773
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 89%●distinct values known / distinct values provided: 83%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 83% ●values known / values provided: 89%●distinct values known / distinct values provided: 83%
Values
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,2,1] => [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
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,1,1,1,2,1] => [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
[1,1,1,1,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [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
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,1,3,1] => [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
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2,1,2] => [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),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,4,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,1,1,2,1] => [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),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1,1,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[1,1,1,1,1,2,1] => [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
[1,1,1,1,1,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[1,1,1,1,2,1,1] => [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
[1,1,1,1,2,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1
[1,1,1,1,3,1] => [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
[1,1,1,1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,2,1,1,1] => [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
[1,1,1,2,1,2] => [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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,2,2,1] => [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)
=> 1
[1,1,1,2,3] => [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
[1,1,1,3,1,1] => [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
[1,1,1,3,2] => [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
[1,1,1,4,1] => [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
[1,1,1,5] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2,1,1,2] => [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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,1,2,1] => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,2,1,3] => [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),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,2,2,1,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,3,1] => [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),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,2,4] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,3,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,2] => [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),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,3,2,1] => [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),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,1,1,1,2] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,1,1,1,2,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ?
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,1,1,1,1,2,1,2] => [5,1,1,1] => ([(0,5),(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
[1,1,1,1,2,1,1,2] => [4,1,2,1] => ([(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)
=> ?
=> ? = 1
[1,1,1,1,2,2,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
[1,1,1,2,1,1,1,2] => [3,1,3,1] => ([(0,7),(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)
=> ?
=> ? = 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => ([(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)
=> ?
=> ? = 2
[1,1,1,2,2,1,1,1] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => ([(0,6),(0,7),(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
[1,1,2,1,1,2,1,1] => [2,1,2,1,2] => ([(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
[1,2,1,1,1,2,1,1] => [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
[2,1,1,1,1,2,1,1] => [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
[2,1,1,1,2,1,1,1] => [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
[1,1,1,1,2,2,2,2] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,3,3,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,3,1,1,3] => [4,1,2,1] => ([(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)
=> ?
=> ? = 1
[1,1,2,2,1,1,2,2] => [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)
=> ?
=> ? = 1
[1,1,2,2,2,2,1,1] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,2,2,1,1,2] => [2,3,2,1] => ([(0,7),(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)
=> ?
=> ? = 1
[1,1,2,1,1,2,2,2] => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,2,1,1,1,2,3] => [2,1,3,1,1] => ([(0,6),(0,7),(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
[1,1,2,1,2,3,1,1] => [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)
=> ?
=> ? = 5
[1,1,2,1,2,2,1,2] => [2,1,1,2,1,1] => ([(0,6),(0,7),(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)
=> ?
=> ? = 2
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => ([(0,7),(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
[1,1,3,2,1,2,1,1] => [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)
=> ?
=> ? = 5
[1,1,3,2,1,1,1,2] => [2,1,1,3,1] => ([(0,7),(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)
=> ?
=> ? = 1
[1,1,3,1,1,3,1,1] => [2,1,2,1,2] => ([(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
[1,1,3,1,1,2,1,2] => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(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)
=> ?
=> ? = 3
[2,2,1,1,1,1,2,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[2,2,1,1,2,2,1,1] => [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)
=> ?
=> ? = 1
[2,2,1,1,2,1,1,2] => [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)
=> ?
=> ? = 1
[2,2,2,2,1,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[2,2,2,1,1,2,1,1] => [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)
=> ?
=> ? = 2
[2,2,2,1,1,1,1,2] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[2,1,1,2,1,1,2,2] => [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
[2,1,1,2,2,2,1,1] => [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
[2,1,1,2,2,1,1,2] => [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)
=> ?
=> ? = 1
[2,1,1,1,1,2,2,2] => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1
[2,1,1,1,2,3,1,1] => [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)
=> ?
=> ? = 3
[2,1,1,1,2,2,1,2] => [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)
=> ?
=> ? = 2
[2,1,1,1,2,1,1,3] => [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)
=> ?
=> ? = 1
[2,1,2,2,1,2,1,1] => [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)
=> ?
=> ? = 3
[2,1,2,2,1,1,1,2] => [1,1,2,3,1] => ([(0,7),(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)
=> ?
=> ? = 1
[2,1,2,1,1,3,1,1] => [1,1,1,2,1,2] => ([(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
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St001135
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 88%●distinct values known / distinct values provided: 83%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 88%●distinct values known / distinct values provided: 83%
Values
[1,1,1,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,1,2] => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,2,2] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,4] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,4,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[2,2,1,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,2,2,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[1,1,1,1,1,3] => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,2,2] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,5] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,4] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,1,1,1,1,1,2] => [7,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,2,1,2] => [5,1,1,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,2,1,2,1] => [4,1,1,1,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,2,2,1,1] => [4,2,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,2,2,1,1,1] => [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,2,1,1,2,1,1] => [2,1,2,1,2] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,2,1,1,1,2,1,1] => [1,1,3,1,2] => [2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[2,1,1,1,1,2,1,1] => [1,4,1,2] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[2,1,1,1,2,1,1,1] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,2,2,2] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,2,1,2,3] => [4,1,1,1,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,3,3,1,1] => [4,2,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,3,2,1,2] => [4,1,1,1,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,3,1,1,3] => [4,1,2,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,2,2,1,1,2,2] => [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,2,2,2,2,1,1] => [2,4,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,2,2,2,1,1,2] => [2,3,2,1] => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,2,1,1,2,2,2] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,2,1,1,1,2,3] => [2,1,3,1,1] => [1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,2,1,2,3,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,2,1,2,2,1,2] => [2,1,1,2,1,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,3,2,1,2,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,3,2,1,1,1,2] => [2,1,1,3,1] => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,3,1,1,3,1,1] => [2,1,2,1,2] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,3,1,1,2,1,2] => [2,1,2,1,1,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[2,2,1,1,1,1,2,2] => [2,4,2] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,2,1,1,2,2,1,1] => [2,2,2,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[2,2,1,1,2,1,1,2] => [2,2,1,2,1] => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,2,2,2,1,1,1,1] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,2,2,1,1,2,1,1] => [3,2,1,2] => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
[2,2,2,1,1,1,1,2] => [3,4,1] => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,1,2,1,1,2,2] => [1,2,1,2,2] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,1,1,2,2,2,1,1] => [1,2,3,2] => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[2,1,1,2,2,1,1,2] => [1,2,2,2,1] => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[2,1,1,1,1,2,2,2] => [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,1,1,2,3,1,1] => [1,3,1,1,2] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[2,1,1,1,2,2,1,2] => [1,3,2,1,1] => [1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 2
[2,1,1,1,2,1,1,3] => [1,3,1,2,1] => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001368
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001368: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001368: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,1,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,1,1,1,2] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[1,1,1,1,1,2,1] => [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)
=> 2
[1,1,1,1,1,3] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,1,2,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,1,3,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,2,1,2] => [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)
=> 3
[1,1,1,2,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 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)
=> 2
[1,1,1,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,5] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,2,1,1,2] => [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)
=> 1
[1,1,2,1,2,1] => [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)
=> 4
[1,1,2,1,3] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,2,1,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,4] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,1,1,1,2] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,2,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,2,1,2] => [5,1,1,1] => ([(0,5),(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
[1,1,1,1,2,1,1,2] => [4,1,2,1] => ([(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)
=> ? = 1
[1,1,1,1,2,1,2,1] => [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)
=> ? = 4
[1,1,1,1,2,2,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,2,1,1,1,2] => [3,1,3,1] => ([(0,7),(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)
=> ? = 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => ([(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)
=> ? = 2
[1,1,1,2,2,1,1,1] => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => ([(0,6),(0,7),(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
[1,1,2,1,1,2,1,1] => [2,1,2,1,2] => ([(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
[1,1,2,1,2,1,2] => [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)
=> ? = 5
[1,2,1,1,1,2,1,1] => [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
[1,2,1,1,2,1,2] => [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)
=> ? = 3
[2,1,1,1,1,2,1,1] => [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
[2,1,1,1,2,1,1,1] => [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
[2,1,1,2,1,2,1] => [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)
=> ? = 4
[2,1,2,1,1,2,1] => [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)
=> ? = 2
[1,1,1,1,2,2,2,2] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,2,1,2,3] => [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)
=> ? = 4
[1,1,1,1,3,3,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,3,2,1,2] => [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)
=> ? = 4
[1,1,1,1,3,1,1,3] => [4,1,2,1] => ([(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)
=> ? = 1
[1,1,2,2,1,1,2,2] => [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)
=> ? = 1
[1,1,2,2,2,2,1,1] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,2,2,1,1,2] => [2,3,2,1] => ([(0,7),(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)
=> ? = 1
[1,1,2,1,1,2,2,2] => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,1,1,1,2,3] => [2,1,3,1,1] => ([(0,6),(0,7),(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
[1,1,2,1,2,3,1,1] => [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)
=> ? = 5
[1,1,2,1,2,2,1,2] => [2,1,1,2,1,1] => ([(0,6),(0,7),(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)
=> ? = 2
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => ([(0,7),(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
[1,1,3,2,1,2,1,1] => [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)
=> ? = 5
[1,1,3,2,1,1,1,2] => [2,1,1,3,1] => ([(0,7),(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)
=> ? = 1
[1,1,3,1,1,3,1,1] => [2,1,2,1,2] => ([(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
[1,1,3,1,1,2,1,2] => [2,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(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)
=> ? = 3
[2,2,1,1,1,1,2,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,2,1,1,2,2,1,1] => [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)
=> ? = 1
[2,2,1,1,2,1,1,2] => [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)
=> ? = 1
[2,2,2,2,1,1,1,1] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[2,2,2,1,1,2,1,1] => [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)
=> ? = 2
[2,2,2,1,1,1,1,2] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,1,1,2,1,1,2,2] => [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
[2,1,1,2,2,2,1,1] => [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
[2,1,1,2,2,1,1,2] => [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)
=> ? = 1
Description
The number of vertices of maximal degree in a graph.
Matching statistic: St001316
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001316: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 82%●distinct values known / distinct values provided: 67%
Values
[1,1,1,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,3] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,1,2] => [5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,4] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,3] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => ([(5,6)],7)
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,1,1,3] => [5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
[1,1,1,1,2,2] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,4] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,1,1,2,1,1,1] => [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
[1,1,1,2,1,2] => [3,1,1,1] => [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)
=> 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,5] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1,1,1,1] => [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
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [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)
=> 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,4] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,1,1,1,2] => [7,1] => [1,7] => ([(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,8] => ([(7,8)],9)
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => [1,7] => ([(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,2,1,2] => [5,1,1,1] => [1,5,1,1] => ([(0,6),(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
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [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)
=> ? = 1
[1,1,1,1,2,1,2,1] => [4,1,1,1,1] => [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)
=> ? = 4
[1,1,1,1,2,2,1,1] => [4,2,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,2,1,1,1,2] => [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)
=> ? = 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [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
[1,1,1,2,2,1,1,1] => [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)
=> ? = 1
[1,1,2,1,1,1,2,1] => [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
[1,1,2,1,1,2,1,1] => [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
[1,1,2,1,2,1,2] => [2,1,1,1,1,1] => [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)
=> ? = 5
[1,2,1,1,1,2,1,1] => [1,1,3,1,2] => [2,1,1,3,1] => ([(0,7),(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
[1,2,1,1,2,1,2] => [1,1,2,1,1,1] => [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)
=> ? = 3
[2,1,1,1,1,2,1,1] => [1,4,1,2] => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[2,1,1,1,2,1,1,1] => [1,3,1,3] => [3,1,3,1] => ([(0,7),(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)
=> ? = 2
[2,1,1,2,1,2,1] => [1,2,1,1,1,1] => [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)
=> ? = 4
[2,1,2,1,1,2,1] => [1,1,1,2,1,1] => [1,1,1,1,2,1] => ([(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)
=> ? = 2
[1,1,1,1,2,2,2,2] => [4,4] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,2,1,2,3] => [4,1,1,1,1] => [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)
=> ? = 4
[1,1,1,1,3,3,1,1] => [4,2,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,3,2,1,2] => [4,1,1,1,1] => [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)
=> ? = 4
[1,1,1,1,3,1,1,3] => [4,1,2,1] => [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)
=> ? = 1
[1,1,2,2,1,1,2,2] => [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)
=> ? = 1
[1,1,2,2,2,2,1,1] => [2,4,2] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,2,2,2,1,1,2] => [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
[1,1,2,1,1,2,2,2] => [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)
=> ? = 1
[1,1,2,1,1,1,2,3] => [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
[1,1,2,1,2,3,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(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)
=> ? = 5
[1,1,2,1,2,2,1,2] => [2,1,1,2,1,1] => [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)
=> ? = 2
[1,1,2,1,2,1,1,3] => [2,1,1,1,2,1] => [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
[1,1,3,2,1,2,1,1] => [2,1,1,1,1,2] => [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(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)
=> ? = 5
[1,1,3,2,1,1,1,2] => [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)
=> ? = 1
[1,1,3,1,1,3,1,1] => [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
[1,1,3,1,1,2,1,2] => [2,1,2,1,1,1] => [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)
=> ? = 3
[2,2,1,1,1,1,2,2] => [2,4,2] => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,2,1,1,2,2,1,1] => [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)
=> ? = 1
[2,2,1,1,2,1,1,2] => [2,2,1,2,1] => [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)
=> ? = 1
[2,2,2,2,1,1,1,1] => [4,4] => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[2,2,2,1,1,2,1,1] => [3,2,1,2] => [2,3,2,1] => ([(0,7),(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
[2,2,2,1,1,1,1,2] => [3,4,1] => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,1,1,2,1,1,2,2] => [1,2,1,2,2] => [2,1,2,1,2] => ([(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)
=> ? = 1
[2,1,1,2,2,2,1,1] => [1,2,3,2] => [2,1,2,3] => ([(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,1,1,2,2,1,1,2] => [1,2,2,2,1] => [1,1,2,2,2] => ([(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)
=> ? = 1
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St000286
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
[1,1,1,2] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,3] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,1,2] => [5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,3] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,4] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,2,3] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,1,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,1] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => ([(5,6)],7)
=> ? = 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,1,3] => [5,1] => [1,5] => ([(4,5)],6)
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,2,2] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,4] => [4,1] => [1,4] => ([(3,4)],5)
=> 1
[1,1,1,2,1,1,1] => [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
[1,1,1,2,1,2] => [3,1,1,1] => [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)
=> 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,5] => [3,1] => [1,3] => ([(2,3)],4)
=> 1
[1,1,2,1,1,1,1] => [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
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [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)
=> 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,2,4] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,4,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,4,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,5,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,1,2] => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,1,1,2] => [7,1] => [1,7] => ([(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,1,3] => [6,1] => [1,6] => ([(5,6)],7)
=> ? = 1
[1,1,1,1,1,2,2] => [5,2] => [2,5] => ([(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,1,3,1] => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,2,1,2] => [4,1,1,1] => [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)
=> ? = 3
[1,1,1,1,2,2,1] => [4,2,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,3,1,1] => [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,2,1,1,2] => [3,1,2,1] => [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)
=> ? = 1
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [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)
=> ? = 4
[1,1,1,2,2,1,1] => [3,2,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,3,1,1,1] => [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
[1,1,2,1,1,1,2] => [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)
=> ? = 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [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)
=> ? = 2
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [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)
=> ? = 4
[1,1,2,2,1,1,1] => [2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,3,1,1,1,1] => [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
[1,2,1,1,1,1,2] => [1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,1,1,1,2,1] => [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
[1,2,1,1,2,1,1] => [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
[2,1,1,1,1,1,2] => [1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,1,1,1,2,1] => [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)
=> ? = 2
[2,1,1,1,2,1,1] => [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)
=> ? = 2
[2,1,1,2,1,1,1] => [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)
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,8] => ([(7,8)],9)
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => [1,7] => ([(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,1,1,4] => [6,1] => [1,6] => ([(5,6)],7)
=> ? = 1
[1,1,1,1,1,2,1,2] => [5,1,1,1] => [1,5,1,1] => ([(0,6),(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
[1,1,1,1,1,2,3] => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,1,3,2] => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,1,4,1] => [5,1,1] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [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)
=> ? = 1
[1,1,1,1,2,1,2,1] => [4,1,1,1,1] => [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)
=> ? = 4
[1,1,1,1,2,1,3] => [4,1,1,1] => [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)
=> ? = 3
[1,1,1,1,2,2,1,1] => [4,2,2] => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,2,2,2] => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? = 1
[1,1,1,1,2,3,1] => [4,1,1,1] => [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)
=> ? = 3
[1,1,1,1,3,1,1,1] => [4,1,3] => [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[1,1,1,1,3,1,2] => [4,1,1,1] => [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)
=> ? = 3
[1,1,1,1,3,2,1] => [4,1,1,1] => [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)
=> ? = 3
[1,1,1,1,4,1,1] => [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,1,1,2,1,1,1,2] => [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)
=> ? = 1
Description
The number of connected components of the complement of a graph.
The complement of a graph is the graph on the same vertex set with complementary edges.
Matching statistic: St001201
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 67%
Values
[1,1,1,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,1,2] => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,2,2] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,4] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,4,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,1,1,1,2] => [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[2,2,1,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,2,2,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,3] => [5,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,2] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,1,2,1,2] => [3,1,1,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,2,2,1] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,2,3] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,1,1] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,3,2] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,4,1] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,1,5] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,1,2,1,1,2] => [2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1,3] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2,1,1] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,4] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,3,2,1] => [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,4,1,1] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,4,2] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,5,1] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,2,1,1,1,2] => [1,1,3,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1,1,1,1,2] => [7,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,3] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,2,2] => [5,2] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,3,1] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,1,1,1] => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,2,2,1] => [4,2,1] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,3,1,1] => [4,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,2,2,1,1] => [3,2,2] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,3,1,1,1] => [3,1,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,2,2,1,1,1] => [2,2,3] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,1,3,1,1,1,1] => [2,1,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,2,1,1,1,1,2] => [1,1,4,1] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[2,1,1,1,1,1,2] => [1,5,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,1,1,1,1,2,1] => [1,4,1,1] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[2,1,1,1,2,1,1] => [1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,3] => [7,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,1,4] => [6,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,2,1,2] => [5,1,1,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,2,3] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,3,2] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,1,4,1] => [5,1,1] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,2,1,2,1] => [4,1,1,1,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,2,1,3] => [4,1,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,2,2,1,1] => [4,2,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
[1,1,1,1,2,2,2] => [4,3] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,2,3,1] => [4,1,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,3,1,1,1] => [4,1,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,3,1,2] => [4,1,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,3,2,1] => [4,1,1,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,4,1,1] => [4,1,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
Description
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000141The maximum drop size of a permutation. St000451The length of the longest pattern of the form k 1 2. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001417The length of a longest palindromic subword of a binary word. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000665The number of rafts of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000871The number of very big ascents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000834The number of right outer peaks of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001571The Cartan determinant of the integer partition.
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