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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> []
=> 0
([],2)
=> [2,2]
=> [2]
=> 1
([(0,1)],2)
=> [3]
=> []
=> 0
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> 3
([(1,2)],3)
=> [6]
=> []
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 1
([(2,3)],4)
=> [6,6]
=> [6]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> 2
Description
The length of the partition.
Matching statistic: St000473
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> []
=> 0
([],2)
=> [2,2]
=> [2]
=> 1
([(0,1)],2)
=> [3]
=> []
=> 0
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> 3
([(1,2)],3)
=> [6]
=> []
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 1
([(2,3)],4)
=> [6,6]
=> [6]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> 2
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> []
=> []
=> 0
([],2)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,1)],2)
=> [3]
=> []
=> []
=> 0
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,2)],3)
=> [6]
=> []
=> []
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> []
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3)],4)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> []
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> []
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> 6
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> []
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> [2,2,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000146
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> []
=> 0
([],2)
=> [2,2]
=> [2]
=> 1
([(0,1)],2)
=> [3]
=> []
=> 0
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> 3
([(1,2)],3)
=> [6]
=> []
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 1
([(2,3)],4)
=> [6,6]
=> [6]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> 2
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,3),(0,4),(0,5),(4,6),(5,6),(6,1),(6,2)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,6),(1,5),(2,5),(3,6),(4,6),(5,3),(5,4)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,3),(1,5),(1,6),(2,5),(2,6),(4,3),(5,4),(6,4)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(4,1),(4,2)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,6),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> ? = 5
Description
The Andrews-Garvan crank of a partition.
If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by
$$
c(\pi) =
\begin{cases}
l(\pi) &\text{if \(\omega(\pi)=0\)}\\
\mu(\pi) - \omega(\pi) &\text{otherwise}.
\end{cases}
$$
This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$
Matching statistic: St000143
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [2]
=> []
=> []
=> 0
([],2)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,1)],2)
=> [3]
=> []
=> []
=> 0
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,2)],3)
=> [6]
=> []
=> []
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> []
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3)],4)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> []
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> []
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> []
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? = 3
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> [2,2,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> ? = 2
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [3]
=> [1,1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> ? = 2
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? = 3
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> [5,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? = 3
([(0,5),(1,4),(4,2),(5,3)],6)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [5,3,2,2,2,2]
=> [3,2,2,2,2]
=> [5,5,1]
=> ? = 5
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [7,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [8,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,3),(0,4),(0,5),(4,6),(5,6),(6,1),(6,2)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,1),(0,2),(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [4,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,6),(1,6),(2,6),(6,3),(6,4),(6,5)],7)
=> [4,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7)
=> [8,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,6),(1,5),(2,5),(3,6),(4,6),(5,3),(5,4)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(5,4),(6,3)],7)
=> [4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> ? = 5
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [8,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,5),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [8,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,6),(1,6),(2,3),(2,6),(4,5),(6,4)],7)
=> [7,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 2
([(0,3),(1,5),(1,6),(2,5),(2,6),(4,3),(5,4),(6,4)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,6),(6,4)],7)
=> [7,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 2
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(4,1),(4,2)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> [8,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,6),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,5),(0,6),(1,2),(1,3),(1,5),(2,6),(3,6),(6,4)],7)
=> [7,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 2
([(0,5),(5,4),(5,6),(6,1),(6,2),(6,3)],7)
=> [8,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,3),(1,2),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [4,3,3,2,2,2]
=> [3,3,2,2,2]
=> [5,5,2]
=> ? = 5
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [5,4,2,2,2,2]
=> [4,2,2,2,2]
=> [5,5,1,1]
=> ? = 5
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(4,3),(5,3),(6,3)],7)
=> [4,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? = 6
([(0,5),(0,6),(1,4),(2,6),(3,6),(4,2),(4,3),(4,5)],7)
=> [7,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 2
([(0,3),(1,4),(1,5),(1,6),(2,6),(3,2),(3,4),(3,5)],7)
=> [7,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 2
([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> [5,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? = 3
([(0,6),(1,4),(4,5),(5,2),(5,3),(5,6)],7)
=> [7,6,5]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 2
([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> ? = 2
([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> [5,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? = 3
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 1 = 0 + 1
([],2)
=> [2,2]
=> 2 = 1 + 1
([(0,1)],2)
=> [3]
=> 1 = 0 + 1
([],3)
=> [2,2,2,2]
=> 4 = 3 + 1
([(1,2)],3)
=> [6]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> [4]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> 2 = 1 + 1
([(2,3)],4)
=> [6,6]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 2 = 1 + 1
([(1,2),(2,3)],4)
=> [4,4]
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 4 = 3 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 4 = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 5 = 4 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> ? = 2 + 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> 2 = 1 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 2 = 1 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> ? = 2 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> ? = 2 + 1
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> [10,4,4]
=> ? = 2 + 1
([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [10,4,4]
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [6,5,3,2,2]
=> ? = 4 + 1
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(1,5),(5,4)],6)
=> [10,4,4]
=> ? = 2 + 1
([(1,3),(1,5),(4,2),(5,4)],6)
=> [10,4,4]
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> [10,10]
=> ? = 1 + 1
([(1,5),(2,3),(3,4),(3,5)],6)
=> [10,10]
=> ? = 1 + 1
([(1,5),(2,3),(3,4),(4,5)],6)
=> [10,4,4]
=> ? = 2 + 1
([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> [11,7,3]
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [6,5,3,2,2]
=> ? = 4 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [7,6,6]
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> [8,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,6),(5,6)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> [8,6,4,2]
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> [8,6,4,2]
=> ? = 3 + 1
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,5)],7)
=> [8,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,6),(1,6),(2,4),(2,6),(4,5),(6,3),(6,5)],7)
=> [8,6,4]
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(2,5),(3,6),(4,5),(5,6)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,4),(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(4,6)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
=> [8,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,5),(1,5),(2,3),(3,6),(5,4),(5,6)],7)
=> [10,6,4]
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6)],7)
=> [6,4,3,3,2]
=> ? = 4 + 1
([(0,5),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> [8,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,6),(1,6),(2,3),(2,6),(4,5),(6,4)],7)
=> [7,6,5]
=> ? = 2 + 1
([(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> [10,10]
=> ? = 1 + 1
([(0,6),(1,5),(1,6),(5,2),(6,3),(6,4)],7)
=> [10,6,4]
=> ? = 2 + 1
([(0,6),(1,4),(1,6),(4,5),(6,2),(6,3),(6,5)],7)
=> [8,6,4]
=> ? = 2 + 1
([(0,6),(1,5),(1,6),(4,2),(5,4),(6,3)],7)
=> [10,4,4]
=> ? = 2 + 1
([(0,5),(0,6),(1,5),(1,6),(5,4),(6,2),(6,3)],7)
=> [6,4,3,3,2]
=> ? = 4 + 1
([(0,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> [8,6,4,2]
=> ? = 3 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,6),(6,4)],7)
=> [7,6,5]
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,6),(4,6),(5,6)],7)
=> [8,6,4,2]
=> ? = 3 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(5,4)],7)
=> [8,6,4]
=> ? = 2 + 1
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,6),(4,5)],7)
=> [10,6,4]
=> ? = 2 + 1
([(0,4),(0,6),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,6),(1,2),(1,3),(1,6),(2,5),(3,5),(5,4),(6,4)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,6),(1,2),(1,3),(1,4),(2,6),(3,6),(4,6),(6,5)],7)
=> [8,2,2,2,2,2,2]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1)],7)
=> [9,6,3]
=> ? = 2 + 1
([(0,3),(0,4),(3,6),(4,5),(4,6),(5,1),(5,2)],7)
=> [9,6,3]
=> ? = 2 + 1
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> [10,10]
=> ? = 1 + 1
([(0,3),(0,6),(1,4),(3,5),(4,2),(4,5),(4,6)],7)
=> [10,6,4]
=> ? = 2 + 1
([(0,2),(0,4),(2,5),(2,6),(3,1),(4,3),(4,5),(4,6)],7)
=> [9,6,3]
=> ? = 2 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000378
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [2]
=> [1,1]
=> 1 = 0 + 1
([],2)
=> [2,2]
=> [4]
=> 2 = 1 + 1
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
([],3)
=> [2,2,2,2]
=> [5,1,1,1]
=> 4 = 3 + 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [5]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [5]
=> 2 = 1 + 1
([(2,3)],4)
=> [6,6]
=> [12]
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [5,1,1,1,1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
([(1,2),(2,3)],4)
=> [4,4]
=> [8]
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [5,1,1,1,1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [2,2,2,2,1]
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2,2,1]
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,4]
=> ? = 6 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [13]
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [5,1,1,1,1,1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [5,5]
=> 4 = 3 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [5,1,1,1,1,1,1]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [8,4]
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [2,2,2,2,2]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [9]
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [6,1,1]
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [9,1,1,1,1,1,1,1]
=> ? = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [8,4]
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [5,5]
=> 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [8,4]
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [6,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,4]
=> ? = 6 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [5,5]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [9,6]
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [6,1,1,1,1,1,1,1]
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [6,6]
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [6,5]
=> 5 = 4 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [5,1,1,1,1,1,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [11,1,1,1]
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [13]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [2,2,2,2,2,2,2,2,2]
=> 3 = 2 + 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> 2 = 1 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [9]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [7,2,2,2,1]
=> ? = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [11,1,1,1]
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [11,1,1,1,1]
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> ?
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [6,2,2,2,1,1,1]
=> ? = 4 + 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [9,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [7,6,1,1,1]
=> ? = 5 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [11,1,1,1,1]
=> ? = 2 + 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 2 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [7,6,1,1,1]
=> ? = 5 + 1
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> ? = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> [3,3,3,2,1,1,1,1]
=> ? = 3 + 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> [3,3,3,1,1,1,1]
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> [10,2,2]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [11,1,1,1,1]
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> ? = 2 + 1
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2]
=> [3,3,3,1,1,1,1]
=> ? = 3 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [11,5]
=> [2,2,2,2,2,1,1,1,1,1,1]
=> ? = 1 + 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 1 + 1
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> [10,4,4]
=> [9,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [9,1,1,1,1,1,1,1,1]
=> ? = 3 + 1
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [9,4]
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> ? = 2 + 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [6,4,3]
=> [9,1,1,1,1]
=> ? = 2 + 1
([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [10,4,4]
=> [9,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> [10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> ? = 1 + 1
([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
=> [7,2,2,2]
=> [3,3,3,1,1,1,1]
=> ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,2,2,2,2]
=> [7,6,1,1,1]
=> ? = 5 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(4,5)],6)
=> [10,2,2,2]
=> [3,3,3,1,1,1,1,1,1,1]
=> ? = 3 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [7,2,2,2,2]
=> [6,2,2,2,1,1,1]
=> ? = 4 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
=> [10,2,2]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [6,5,3,2,2]
=> ?
=> ? = 4 + 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000445
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 86%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 86%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 3
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 3
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> 2
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
=> ? = 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 3
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ?
=> ? = 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ?
=> ? = 4
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 2
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 3
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0,0]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ?
=> ? = 3
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> [10,6]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
=> ? = 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ?
=> ? = 2
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [11,5]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ?
=> ? = 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 3
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 2
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 2
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 71%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 71%
Values
([],1)
=> [2]
=> 100 => 1 = 0 + 1
([],2)
=> [2,2]
=> 1100 => 2 = 1 + 1
([(0,1)],2)
=> [3]
=> 1000 => 1 = 0 + 1
([],3)
=> [2,2,2,2]
=> 111100 => 4 = 3 + 1
([(1,2)],3)
=> [6]
=> 1000000 => 1 = 0 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> 10100 => 2 = 1 + 1
([(0,2),(2,1)],3)
=> [4]
=> 10000 => 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> 10100 => 2 = 1 + 1
([(2,3)],4)
=> [6,6]
=> 11000000 => 2 = 1 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> 100001100 => 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1011100 => 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 10000000 => 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 2 = 1 + 1
([(1,2),(2,3)],4)
=> [4,4]
=> 110000 => 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 2 = 1 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> 100001100 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1011100 => 4 = 3 + 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111000 => 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1001000 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 101100 => 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 100000 => 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 10000000 => 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> 1000011111100 => ? = 6 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 101000000 => 2 = 1 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 1000001100 => ? = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 10011100 => 4 = 3 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> 100000000000000 => ? = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> 1000001100 => ? = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 11001100 => 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1000100 => 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 1011000 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 1010000 => 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 1001100 => 3 = 2 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 11110000 => 4 = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 1001100 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> 1000011111100 => ? = 6 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 4 = 3 + 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> 1000111000 => ? = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> 10000010100 => ? = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 100101100 => 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 10111100 => 5 = 4 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1000100 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 1000001100 => ? = 2 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 101001000 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 101000000 => 2 = 1 + 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> 111000000 => 3 = 2 + 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> 100001000000 => ? = 1 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> 1000001000 => ? = 1 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 1010000 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> 10000111100 => ? = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 100000100 => 2 = 1 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 1001100 => 3 = 2 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> 10000000000 => 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> 1000111000 => ? = 3 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> 101001000 => 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 1010000 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> 100000000100 => ? = 1 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> 100000000 => 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 1000001100 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> 10000010100 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 100101100 => 4 = 3 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> 1000001000 => ? = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> 100000000000000 => ? = 0 + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> 1000000000000 => ? = 0 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> 1001011000 => ? = 3 + 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> 100111100 => ? = 4 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> 10000001100 => ? = 2 + 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> 1011000000 => ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> 100010000000 => ? = 1 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> 100000111100 => ? = 4 + 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> 1000000001100 => ? = 2 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> 1000000110000 => ? = 2 + 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> 1100111100 => ? = 5 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [11]
=> 100000000000 => ? = 0 + 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> 1001011000 => ? = 3 + 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> 100111100 => ? = 4 + 1
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> 110000000000 => ? = 1 + 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> 1000000001100 => ? = 2 + 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> 1100111100 => ? = 5 + 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> 100111100 => ? = 4 + 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> 100111100 => ? = 4 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> 100000101100 => ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> 10000011100 => ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> 100111100 => ? = 4 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> 1000000001100 => ? = 2 + 1
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> 1000000001100 => ? = 2 + 1
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> 10000001100 => ? = 2 + 1
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> [10,2,2]
=> 1000000001100 => ? = 2 + 1
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> [10,6]
=> 100001000000 => ? = 1 + 1
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2]
=> 10000011100 => ? = 3 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [11,5]
=> 1000000100000 => ? = 1 + 1
([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> 1000000000000000 => ? = 0 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> 10000001100 => ? = 2 + 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> 1001000000 => ? = 1 + 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> 1001011000 => ? = 3 + 1
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> 10000001100 => ? = 2 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000307
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
([],1)
=> 1 = 0 + 1
([],2)
=> 2 = 1 + 1
([(0,1)],2)
=> 1 = 0 + 1
([],3)
=> 4 = 3 + 1
([(1,2)],3)
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,3)],4)
=> 2 = 1 + 1
([(1,2),(1,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> 4 = 3 + 1
([(0,2),(0,3),(3,1)],4)
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(1,2),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> 3 = 2 + 1
([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> 7 = 6 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,3),(1,4),(4,2)],5)
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(2,3),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(4,2),(4,3)],5)
=> 4 = 3 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> 7 = 6 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,3)],5)
=> 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 5 = 4 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ? = 3 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ? = 1 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ? = 2 + 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ? = 2 + 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 2 + 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> ? = 3 + 1
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ? = 2 + 1
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> ? = 2 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> ? = 1 + 1
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(4,5)],6)
=> ? = 3 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 4 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 1
([(0,5),(1,2),(1,5),(5,3),(5,4)],6)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ? = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,5),(1,2),(1,3),(1,5),(5,4)],6)
=> ? = 2 + 1
([(0,3),(1,2),(1,4),(1,5),(3,4),(3,5)],6)
=> ? = 3 + 1
([(0,5),(1,2),(1,3),(3,5),(5,4)],6)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> ? = 1 + 1
([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ? = 1 + 1
([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ? = 1 + 1
([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(4,3),(5,3)],6)
=> ? = 2 + 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3)],6)
=> ? = 2 + 1
([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
([(0,4),(0,5),(1,3),(1,5),(5,2)],6)
=> ? = 2 + 1
([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 1
([(0,2),(0,4),(1,3),(1,4),(3,5),(4,5)],6)
=> ? = 1 + 1
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> ? = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ? = 1 + 1
([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> ? = 1 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000052The number of valleys of a Dyck path not on the x-axis. St000734The last entry in the first row of a standard tableau. St000157The number of descents of a standard tableau. St000733The row containing the largest entry of a standard tableau. St000507The number of ascents of a standard tableau. St000053The number of valleys of the Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001712The number of natural descents of a standard Young tableau. St000015The number of peaks of a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001596The number of two-by-two squares inside a skew partition. St001354The number of series nodes in the modular decomposition of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1).
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