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Your data matches 179 different statistics following compositions of up to 3 maps.
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Matching statistic: St001000
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001000: 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]
=> 1
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 6
[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]
=> 3
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 1
[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]
=> 1
[1,1,0,1,1,0,0,0]
=> 4
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 10
[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]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 2
[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]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 3
[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]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 6
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001199
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,3}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[1,0,1,0,1,0]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {1,2,6}
[1,1,0,0,1,0]
=> [2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {1,2,6}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,2,6}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {2,3,10}
[1,1,1,0,0,0,1,0]
=> [3]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {2,3,10}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {2,3,10}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [3]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1]
=> [1,1,0,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [2,2]
=> [1,1,1,0,0,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> [1,0]
=> ? ∊ {2,2,2,2,2,3,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,3,6,15}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000772
(load all 51 compositions to match this statistic)
(load all 51 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> ? = 3
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {2,6}
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ? ∊ {2,6}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,10}
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {2,3,3,4,10}
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,10}
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {2,3,3,4,10}
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? ∊ {2,3,3,4,10}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001571
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 3
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,6}
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,6}
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[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]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 4
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,5,5,6,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 1
Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Matching statistic: St001933
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> []
=> ? = 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? = 3
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,6}
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,6}
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,3,4,10}
[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]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001250
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2] => [2]
=> []
=> ? = 3
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? ∊ {2,6}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {2,6}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {2,3,3,4,10}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {2,3,3,4,10}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {2,3,3,4,10}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {2,3,3,4,10}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {2,3,3,4,10}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St001281
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001281: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> ([],1)
=> ? = 3
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {2,6}
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],1)
=> ? ∊ {2,6}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {2,3,3,3,10}
[1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {2,3,3,3,10}
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {2,3,3,3,10}
[1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {2,3,3,3,10}
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> ? ∊ {2,3,3,3,10}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,4,4,5,5,6,15}
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: St001432
(load all 8 compositions to match this statistic)
(load all 8 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
St001432: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> []
=> ? = 3
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {2,6}
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {2,6}
[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]
=> 2
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,3,3,4,10}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,3,3,4,10}
[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]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,3,3,4,10}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,3,3,4,10}
[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]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {2,3,3,4,10}
[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]
=> 3
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 1
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 2
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 1
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 2
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 3
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 1
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 2
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 1
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 2
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 3
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 1
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 2
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 3
[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]
=> []
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,6,6,15}
[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]
=> 4
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001498
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 3
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {2,6}
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {2,6}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,10}
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,10}
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,10}
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,10}
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,3,4,10}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,2,2,4,5,5,6,6,15}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001914
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001914: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,1] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2] => [2]
=> []
=> ? = 3
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [3] => [3]
=> []
=> ? ∊ {2,6}
[1,1,1,0,0,0]
=> [3] => [3]
=> []
=> ? ∊ {2,6}
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {1,3,3,4,10}
[1,1,0,1,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {1,3,3,4,10}
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [4]
=> []
=> ? ∊ {1,3,3,4,10}
[1,1,1,0,1,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {1,3,3,4,10}
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> []
=> ? ∊ {1,3,3,4,10}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [3,2]
=> [2]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,0,0,1,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,0,1,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,0,1,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,1,0,0,1,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,1,0,1,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
[1,1,1,1,1,0,0,0,0,0]
=> [5] => [5]
=> []
=> ? ∊ {1,2,3,3,4,4,4,4,4,4,5,6,6,15}
Description
The size of the orbit of an integer partition in Bulgarian solitaire.
Bulgarian solitaire is the dynamical system where a move consists of removing the first column of the Ferrers diagram and inserting it as a row.
This statistic returns the number of partitions that can be obtained from the given partition.
The following 169 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000993The multiplicity of the largest part of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000144The pyramid weight of the Dyck path. St000293The number of inversions of a binary word. St000335The difference of lower and upper interactions. St000395The sum of the heights of the peaks of a Dyck path. St000418The number of Dyck paths that are weakly below a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000631The number of distinct palindromic decompositions of a binary word. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000922The minimal number such that all substrings of this length are unique. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001809The index of the step at the first peak of maximal height in a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001959The product of the heights of the peaks of a Dyck path. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000327The number of cover relations in a poset. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000937The number of positive values of the symmetric group character corresponding to the partition. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000260The radius of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001128The exponens consonantiae of a partition. St000420The number of Dyck paths that are weakly above a Dyck path. St000438The position of the last up step in a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000981The length of the longest zigzag subpath. 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. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001808The box weight or horizontal decoration of a Dyck path. St001964The interval resolution global dimension of a poset. St001060The distinguishing index of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. 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 $eA$. 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 $eA$. 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 $eA$. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St001943The sum of the squares of the hook lengths of an integer partition. St001645The pebbling number of a connected graph. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000934The 2-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. 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. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001541The Gini index of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000264The girth of a graph, which is not a tree.
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