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Your data matches 135 different statistics following compositions of up to 3 maps.
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Matching statistic: St001507
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St001507: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.
Matching statistic: St000208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2] => [2]
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St001389
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2] => [2]
=> []
=> ? = 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,1,1,1,1,1,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St001199
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 61%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1}
[1,1,0,0]
=> [2] => [1] => [1,0]
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1}
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [3] => [1] => [1,0]
=> ? ∊ {0,1,1}
[1,1,1,0,0,0]
=> [3] => [1] => [1,0]
=> ? ∊ {0,1,1}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,1,1,1,0,0,0,0]
=> [4] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [1] => [1,0]
=> ? ∊ {0,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000832
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000832: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000832: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[1,1,0,0]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5] => ? ∊ {0,2,2,2,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,4,2,3,5,6] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3] => ? ∊ {0,2,2,2,3,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,3,1,4,5,6] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7] => ? ∊ {0,2,2,2,3,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7] => ? ∊ {0,2,2,2,3,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5] => ? ∊ {0,2,2,2,3,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,4,6,1,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,4,6,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2,2,2,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? ∊ {0,2,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,4,2,3,5,8,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3,8,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [2,3,1,4,5,8,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,4,7,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,8,5,9,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,2,3,6,7,5,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,8,3,9,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,6,7,5,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,9,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,4,1,5,6,7,3,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,4,5,7,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,4,2,8,3,5,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [2,3,1,8,4,5,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,2,7,3,4,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,8,9,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,2,6,3,5,7,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,8,9,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,6,4,5,7,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,9,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,5,3,6,7,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,3,1,7,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,6,3,8,5,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,8,5,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5,8,9] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,8,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,1,6,4,7,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,9,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,6,3,7,5,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,6,1,7,8,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,4,5,6,1,7,8,3] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6}
Description
The number of permutations obtained by reversing blocks of three consecutive numbers.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {0,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {0,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {0,1,1,1}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,6}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [5,6,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [2,6,3,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [6,2,4,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => [4,2,5,6,1,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => [4,5,2,6,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [4,5,6,2,1,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,5,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [4,2,6,1,3,5] => ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => [4,6,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => [6,4,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => [5,2,6,1,3,4] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => [5,6,2,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,3,4,6] => [5,2,1,3,4,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [5,2,1,6,3,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => [6,2,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,6,4] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,5,2,6,4] => [5,3,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,5,6,2,4] => [5,6,3,1,2,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => [5,3,1,2,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6} + 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: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 83%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? = 0 + 1
[1,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,1} + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1} + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1} + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1} + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,6,3,2,4,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6,1,3,4,2,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,4,2,5,3] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,6,2,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6,2,1,5,3,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,5,6,1,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [3,1,6,2,5,4] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [2,6,3,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6,2,1,4,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [2,1,6,5,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [5,4,2,6,1,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,6,3,2,5,4] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3} + 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [3,1,5,4,6,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [6,5,2,3,1,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [4,6,5,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [3,4,1,6,5,2] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [5,1,2,3,7,4,6] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [1,7,2,3,4,6,5] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [1,4,2,7,3,5,6] => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [4,1,2,6,7,3,5] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [1,5,2,3,4,7,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,2,7,3,4,6,5] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,6,2,3,5,7,4] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => [7,1,2,4,5,3,6] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => [7,4,1,2,6,3,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [7,1,2,5,6,3,4] => [7,1,5,2,3,6,4] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,3,2,7,4,5,6] => ([(1,2),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => [1,6,3,7,2,4,5] => ([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => [3,5,1,7,2,4,6] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [3,1,7,6,2,4,5] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => [3,1,5,6,7,2,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,1,4,2,3,5,6] => [1,4,2,3,5,7,6] => ([(2,3),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [1,4,2,3,6,7,5] => ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,2,5,3,4,7,6] => ([(2,3),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,2,6,3,4,7,5] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,2,6,3,5,7,4] => ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [5,4,1,2,7,3,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [7,1,4,2,6,3,5] => [4,1,7,2,3,6,5] => ([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,2,5,3,6,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [1,5,2,4,6,7,3] => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6} + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => [3,7,1,4,2,5,6] => ([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => [6,1,3,4,7,2,5] => ([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [3,1,7,5,2,4,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [6,3,1,5,7,2,4] => ([(0,4),(0,6),(1,3),(1,5),(2,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,4,2,5,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [7,1,3,4,5,2,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [7,6,1,3,4,2,5] => ([(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)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,7,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 0
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1,1}
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1,1}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1,1}
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,5,3,4,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000454
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [[1]]
=> [1] => ([],1)
=> 0
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([],2)
=> 0
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3}
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => ([(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3}
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3}
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3}
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,0,1,1,0,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,0,1,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,0,1,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,0,1,0,0,0]
=> [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,0,1,0,0,0,0]
=> [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,0,0,0,0,0]
=> [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,6}
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.
The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001432The order dimension of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001644The dimension of a graph. St000914The sum of the values of the Möbius function of a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000035The number of left outer peaks of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000456The monochromatic index of a connected graph. St001344The neighbouring number of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000862The number of parts of the shifted shape of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001597The Frobenius rank of a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001624The breadth of a lattice. St000632The jump number of the poset. St001413Half the length of the longest even length palindromic prefix of a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000028The number of stack-sorts needed to sort a permutation. St001487The number of inner corners of a skew partition. St001394The genus of a permutation.
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