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Your data matches 86 different statistics following compositions of up to 3 maps.
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Matching statistic: St001027
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001027: 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]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 0
[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]
=> 2
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 2
[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]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000870
(load all 2 compositions to match this statistic)
(load all 2 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
St000870: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2] => [2]
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[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]
=> []
=> ? ∊ {0,2}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[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]
=> 2
[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]
=> []
=> ? ∊ {0,0,1,3,3}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,3,3}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,3,3}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,3,3}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,3,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 2
[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]
=> 2
[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]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 3
[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]
=> 2
[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]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[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]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
[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]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[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]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[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]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,4,4,4}
[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]
=> 5
[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]
=> 4
[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]
=> 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> 3
[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]
=> 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,5,5,5,5,5}
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St000144
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [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
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [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
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [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]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [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]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5}
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St001020
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 86%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [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
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> [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
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> [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]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> [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]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,5,5,5,5,5,5}
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001416
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001416: Binary words ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001416: Binary words ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [[1],[]]
=> []
=> => ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> []
=> => ? ∊ {1,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> => ? ∊ {1,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1000 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1010 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 100 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 100 => 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1010 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1000 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1010 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
Description
The length of a longest palindromic factor of a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.
Matching statistic: St001417
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [[1],[]]
=> []
=> => ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> []
=> => ? ∊ {1,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> => ? ∊ {1,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> => ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> => ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 10 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1000 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> 100 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1010 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> [1]
=> 10 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 110 => 2
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 100 => 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 10 => 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 100 => 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [1]
=> 10 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> => ? ∊ {0,0,0,0,1,2,2,2,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1010 => 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 110 => 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[4,4,1],[3]]
=> [3]
=> 1000 => 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> 100 => 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [2,1]
=> 1010 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,5,5,5,5,5,5}
Description
The length of a longest palindromic subword of a binary word.
A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword.
Any binary word of length $n$ contains a palindromic subword of length at least $n/2$, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with $n/2$ zeros and ending with $n/2$ ones.
Matching statistic: St001515
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1],[]]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [[1,1],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> [[2],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [[3],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> []
=> ? ∊ {0,1,2,2}
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[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]
=> 2
[1,1,0,1,0,1,0,0]
=> [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[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]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[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]
=> 2
[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]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[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]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3,3,3,3,3,4,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[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]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[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],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[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]
=> 2
[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]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[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]
=> 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[5,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 2
[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]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [[5,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [[4,2,2],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000259
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 86%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1,1}
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1,1}
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,1,1}
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [6,1,3,2,4,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [1,4,2,3,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => [5,1,3,2,4,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [4,1,2,5,3,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,5,6,3] => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,5,1,6,3] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,1,3,4,2,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [5,1,3,6,2,4] => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [5,6,1,3,2,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,5,3,4,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,5,1,3,4,6] => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,3,5,4,6] => [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,3,5,6,4] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,3,1,5,6,4] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,5,1,6,4] => [6,1,2,4,3,5] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 3
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,6,4] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,6,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,5,2,6,4] => [6,2,4,1,3,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,5,6,2,4] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000668
(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
St000668: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 48%●distinct values known / distinct values provided: 43%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 48%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> ? ∊ {1,1}
[1,1,0,0]
=> [2] => [2]
=> []
=> ? ∊ {1,1}
[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]
=> ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> ? ∊ {0,1,2,2}
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1,2,2}
[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]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[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]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
(load all 2 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
St000708: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 57%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 0
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> ? ∊ {1,1}
[1,1,0,0]
=> [2] => [2]
=> []
=> ? ∊ {1,1}
[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]
=> ? ∊ {0,1,2,2}
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> ? ∊ {0,1,2,2}
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1,2,2}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {0,1,2,2}
[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]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[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]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[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
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [3,2,1]
=> [2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5}
Description
The product of the parts of an integer partition.
The following 76 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000667The greatest common divisor of the parts of the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. 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. St001933The largest multiplicity of a part in an integer partition. St000993The multiplicity of the largest part of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000260The radius of a connected graph. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000145The Dyson rank of a partition. St000474Dyson's crank of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000770The major index of an integer partition when read from bottom to top. 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. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St000819The propagating number of a perfect matching. St001432The order dimension of the partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001060The distinguishing index of a graph. St000239The number of small weak excedances. St000338The number of pixed points of a permutation. St000873The aix statistic of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000454The largest eigenvalue of a graph if it is integral. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000455The second largest eigenvalue of a graph if it is integral. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001267The length of the Lyndon factorization of the binary word.
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