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Your data matches 52 different statistics following compositions of up to 3 maps.
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Matching statistic: St000761
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000761: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 1
[2,1] => 0
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 0
[2,2] => 0
[3,1] => 0
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 0
[2,1,2] => 1
[2,2,1] => 0
[2,3] => 1
[3,1,1] => 0
[3,2] => 0
[4,1] => 0
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 2
[1,2,2,1] => 1
[1,2,3] => 2
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 0
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The number of ascents in an integer composition.
A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
Matching statistic: St000260
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1] => ([],1)
=> 0
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1] => ([],1)
=> 0
[1,2] => [1,1] => [2] => ([],2)
=> ? ∊ {0,1}
[2,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,1}
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1] => ([],1)
=> 0
[1,1,2] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,1}
[1,3] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,1}
[2,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[2,2] => [2] => [1] => ([],1)
=> 0
[3,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,1}
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1] => ([],1)
=> 0
[1,1,1,2] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,3] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,2] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,1,1}
[1,4] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,1,1}
[2,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[2,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,1,1}
[2,2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[2,3] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,1,1}
[3,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[3,2] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,1,1}
[4,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,1,1}
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [6] => [1] => ([],1)
=> 0
[1,1,1,1,2] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,1,3] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,2] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,3,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,1,4] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,2,2,1] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,3,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,4,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[1,5] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1] => [1,4] => [1,1] => ([(0,1)],2)
=> 1
[2,1,1,2] => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,1,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[2,1,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[2,2,1,1] => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[2,2,2] => [3] => [1] => ([],1)
=> 0
[2,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[2,4] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[3,1,1,1] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[3,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[3,3] => [2] => [1] => ([],1)
=> 0
[4,1,1] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[4,2] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[5,1] => [1,1] => [2] => ([],2)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1] => ([],1)
=> 0
[1,1,1,1,1,2] => [5,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[1,1,1,1,3] => [4,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,1,2,2] => [3,2] => [1,1] => ([(0,1)],2)
=> 1
[1,1,1,3,1] => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[1,1,1,4] => [3,1] => [1,1] => ([(0,1)],2)
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,2,1,2] => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {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}
[1,1,2,2,1] => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,2,3] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[1,1,3,1,1] => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,1,3,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[1,1,4,1] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[1,1,5] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,1,1,1,1] => [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,1,1,2] => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {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}
[1,2,1,3] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {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}
[1,2,2,1,1] => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[1,2,2,2] => [1,3] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {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}
[1,2,4] => [1,1,1] => [3] => ([],3)
=> ? ∊ {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}
[1,3,1,1,1] => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,3,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {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}
[1,3,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {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}
[1,3,3] => [1,2] => [1,1] => ([(0,1)],2)
=> 1
[1,4,1,1] => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,4,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {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}
[1,5,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {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}
[1,6] => [1,1] => [2] => ([],2)
=> ? ∊ {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}
[2,1,1,2,1] => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {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}
[2,1,3,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {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}
[2,1,4] => [1,1,1] => [3] => ([],3)
=> ? ∊ {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}
[2,2,1,2] => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {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}
[2,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {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}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000781
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 47%●distinct values known / distinct values provided: 25%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 47%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 1
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001432
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
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: St001913
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001913: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 1
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
Description
The number of preimages of an integer partition in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
Partitions without preimages are called garden of eden partitions [1].
Matching statistic: St001732
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 75%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [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 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2} + 1
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3} + 1
Description
The number of peaks visible from the left.
This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000897
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000897: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 0
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 0
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 0
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 0
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 0
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 0
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 0
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of different multiplicities of parts of an integer partition.
Matching statistic: St001011
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001212
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 75%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2}
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St000659
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 75%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1}
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,1,1}
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,1,1}
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,1,1,1,1}
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,4,1] => [[4,4,1,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,4] => [[5,2,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[2,5] => [[6,2],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,2,2,2,2,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The number of rises of length at least 2 of a Dyck path.
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001877Number of indecomposable injective modules with projective dimension 2. St000145The Dyson rank of a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000836The number of descents of distance 2 of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001096The size of the overlap set of a permutation. St001323The independence gap of a graph. 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. St001871The number of triconnected components of a graph. St000035The number of left outer peaks of a permutation. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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 $eA$. St000456The monochromatic index of a connected graph. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001960The number of descents of a permutation minus one if its first entry is not one. St001845The number of join irreducibles minus the rank of a lattice. St001651The Frankl number of a lattice. St000031The number of cycles in the cycle decomposition of a permutation.
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