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Your data matches 148 different statistics following compositions of up to 3 maps.
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Matching statistic: St001017
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001017: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 0
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 0
[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]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001280
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 71%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> []
=> ? = 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,3}
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,3}
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,4}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,4}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,4}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,4}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,2,2,2,4}
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,3,3,3,3,3,5}
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 3
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,6}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001281
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 71%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 1
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> 0
[1,1,0,0]
=> [2] => [1] => ([],1)
=> ? = 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [3] => [1] => ([],1)
=> ? ∊ {1,3}
[1,1,1,0,0,0]
=> [3] => [1] => ([],1)
=> ? ∊ {1,3}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => ([],2)
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [1] => ([],1)
=> ? ∊ {2,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [4] => [1] => ([],1)
=> ? ∊ {2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [1] => ([],1)
=> ? ∊ {2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4] => [1] => ([],1)
=> ? ∊ {2,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> [4] => [1] => ([],1)
=> ? ∊ {2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [1] => ([],1)
=> ? ∊ {0,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => ([],6)
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6] => [1] => ([],1)
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,6}
Description
The normalized isoperimetric number of a graph.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$
i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\},
$$
where
$$
\partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}.
$$
This statistic is $i(G)\cdot\lfloor n/2\rfloor$.
Matching statistic: St000681
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 71%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {0,2}
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,1,1,3}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,1,1,3}
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {0,1,1,3}
[1,1,1,0,0,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> 3
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => [3,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,1,4,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,4,4,4,4,4,4,4,4,6}
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000698
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 60%●distinct values known / distinct values provided: 43%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 60%●distinct values known / distinct values provided: 43%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,2}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,3}
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,0,0,1,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,1,1,1,3}
[1,1,0,1,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,1,1,1,3}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [2]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,4,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000714
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 57%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {0,2}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,3}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,3}
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,3}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {0,1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {0,0,0,1,1,1,1,2,2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [2]
=> 3
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> [2,1,1]
=> 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St001124
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 57%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> []
=> []
=> []
=> ? = 1
[1,0,1,0]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,2}
[1,1,0,0]
=> []
=> []
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {1,1,1,3}
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,0,1,0,0]
=> [1]
=> [1,0]
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,2,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,2,2,2,2,4}
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {1,2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> []
=> ? ∊ {1,2,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? ∊ {1,2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,2,1,1,1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [5,4,2,1,1,1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,2,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [7,4,2,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,3,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,2]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,3,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,3,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,3,3,3,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,2,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [7,6,2,2,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [7,6,2,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,3,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,3,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1]
=> 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [7,6,3,3,3,3,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,2,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,2,1,1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> 3
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,1,1,1,1]
=> 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000755
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 47%●distinct values known / distinct values provided: 29%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 47%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,2}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,1,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St001247
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 71%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,2}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 0
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001250
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 71%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 71%
Values
[1,0]
=> [1] => [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,2}
[1,1,0,0]
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,2}
[1,0,1,0,1,0]
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,1,0,0]
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,1,0,0,1,0]
=> [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,1,1,0,0,0]
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,0,1,1,1,0,0,0]
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,1,0,1,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,1,1,0,0,0,1,0]
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,1,1,0,1,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,1,1,1,0,0,0,0]
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [[4,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [[5],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,6}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 3
Description
The number of parts of a partition that are not congruent 0 modulo 3.
The following 138 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000993The multiplicity of the largest part of an integer partition. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000503The maximal difference between two elements in a common block. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000354The number of recoils of a permutation. St000456The monochromatic index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St000137The Grundy value of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001118The acyclic chromatic index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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 $eA$. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000929The constant term of the character polynomial of an integer partition. St001176The size of a partition minus its first part. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001961The sum of the greatest common divisors of all pairs of parts. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000944The 3-degree of an integer partition. St001383The BG-rank of an integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001569The maximal modular displacement of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001060The distinguishing index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000441The number of successions of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000567The sum of the products of all pairs of parts. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001405The number of bonds in a permutation. St001516The number of cyclic bonds of a permutation. St000454The largest eigenvalue of a graph if it is integral.
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