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Your data matches 135 different statistics following compositions of up to 3 maps.
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Matching statistic: St000307
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Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 4
([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([],4)
=> 8
([(2,3)],4)
=> 2
([(1,2),(1,3)],4)
=> 3
([(0,1),(0,2),(0,3)],4)
=> 4
([(0,2),(0,3),(3,1)],4)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> 2
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([],5)
=> 16
([(3,4)],5)
=> 4
([(2,3),(2,4)],5)
=> 6
([(1,2),(1,3),(1,4)],5)
=> 7
([(0,1),(0,2),(0,3),(0,4)],5)
=> 8
([(0,2),(0,3),(0,4),(4,1)],5)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(4,2)],5)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(2,3),(3,4)],5)
=> 4
([(1,4),(4,2),(4,3)],5)
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> 4
([(2,4),(3,4)],5)
=> 6
([(1,4),(2,4),(4,3)],5)
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> 3
([(1,4),(2,4),(3,4)],5)
=> 7
([(0,4),(1,4),(2,4),(4,3)],5)
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> 8
([(0,4),(1,4),(2,3)],5)
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
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 $P \setminus I$.
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
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> [1]
=> 1
([],2)
=> [2,2]
=> [2,2]
=> [2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> [1,1]
=> 1
([],3)
=> [2,2,2,2]
=> [4,4]
=> [4]
=> 4
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> [8]
=> 8
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [2,2,2,2,2]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [3,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [4,1]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [3,1,1,1,1]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [4,1]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> [3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> [3,1]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [16,16]
=> [16]
=> 16
([(3,4)],5)
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> ? = 4
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> 6
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> [7,1,1,1,1]
=> 7
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [8,1]
=> 8
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [4,1,1]
=> 4
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [3,3,1]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 3
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> [4,4,4]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [4,1,1]
=> 4
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> 6
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> [7,1,1,1,1]
=> 7
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [4,1,1]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [8,8,1]
=> [8,1]
=> 8
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [4,4,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [4,2,1,1]
=> 4
Description
The largest part of an integer partition.
Matching statistic: St000010
(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
St000010: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Values
([],1)
=> [2]
=> []
=> 0 = 1 - 1
([],2)
=> [2,2]
=> [2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> []
=> 0 = 1 - 1
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(1,2)],3)
=> [6]
=> []
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> 7 = 8 - 1
([(2,3)],4)
=> [6,6]
=> [6]
=> 1 = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? ∊ {4,16} - 1
([(3,4)],5)
=> [6,6,6,6]
=> [6,6,6]
=> ? ∊ {4,16} - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> 5 = 6 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> 7 = 8 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> 7 = 8 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> 4 = 5 - 1
Description
The length of the 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: 78% ●values known / values provided: 94%●distinct values known / distinct values provided: 78%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000146: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 94%●distinct values known / distinct values provided: 78%
Values
([],1)
=> [2]
=> []
=> 0 = 1 - 1
([],2)
=> [2,2]
=> [2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> []
=> 0 = 1 - 1
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(1,2)],3)
=> [6]
=> []
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 8 - 1
([(2,3)],4)
=> [6,6]
=> [6]
=> 1 = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? ∊ {4,8,8,16} - 1
([(3,4)],5)
=> [6,6,6,6]
=> [6,6,6]
=> ? ∊ {4,8,8,16} - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> 5 = 6 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? ∊ {4,8,8,16} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> 5 = 6 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? ∊ {4,8,8,16} - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> 2 = 3 - 1
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: St001280
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [2]
=> 1
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 1
([],3)
=> [2,2,2,2]
=> 4
([(1,2)],3)
=> [6]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> 8
([(2,3)],4)
=> [6,6]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? ∊ {3,4,6,6,7,7,16}
([(3,4)],5)
=> [6,6,6,6]
=> ? ∊ {3,4,6,6,7,7,16}
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> ? ∊ {3,4,6,6,7,7,16}
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> ? ∊ {3,4,6,6,7,7,16}
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 8
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 4
([(1,3),(1,4),(4,2)],5)
=> [14]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 3
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 4
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> ? ∊ {3,4,6,6,7,7,16}
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> ? ∊ {3,4,6,6,7,7,16}
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> 8
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 5
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 3
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 2
([(1,4),(2,3)],5)
=> [6,6,6]
=> ? ∊ {3,4,6,6,7,7,16}
Description
The number of parts of an integer partition that are at least two.
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: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 92%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [2]
=> []
=> 0 = 1 - 1
([],2)
=> [2,2]
=> [2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> []
=> 0 = 1 - 1
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(1,2)],3)
=> [6]
=> []
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 1 = 2 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? = 8 - 1
([(2,3)],4)
=> [6,6]
=> [6]
=> 1 = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? ∊ {4,6,6,8,8,16} - 1
([(3,4)],5)
=> [6,6,6,6]
=> [6,6,6]
=> ? ∊ {4,6,6,8,8,16} - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> ? ∊ {4,6,6,8,8,16} - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? ∊ {4,6,6,8,8,16} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> ? ∊ {4,6,6,8,8,16} - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> 6 = 7 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> ? ∊ {4,6,6,8,8,16} - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> 3 = 4 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> 4 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> 2 = 3 - 1
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: 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: 56% ●values known / values provided: 87%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 87%●distinct values known / distinct values provided: 56%
Values
([],1)
=> [2]
=> []
=> []
=> 0 = 1 - 1
([],2)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> []
=> []
=> 0 = 1 - 1
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
([(1,2)],3)
=> [6]
=> []
=> []
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> []
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? = 8 - 1
([(2,3)],4)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> []
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> []
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> []
=> 0 = 1 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [15,15]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(3,4)],5)
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> []
=> 0 = 1 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3 = 4 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 3 = 4 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> [2,2,1]
=> 2 = 3 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 3 = 4 - 1
([(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 = 5 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> ? ∊ {3,4,4,6,6,7,7,8,8,16} - 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 4 = 5 - 1
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: St000378
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 85%●distinct values known / distinct values provided: 56%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 85%●distinct values known / distinct values provided: 56%
Values
([],1)
=> [2]
=> [1,1]
=> 1
([],2)
=> [2,2]
=> [4]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1
([],3)
=> [2,2,2,2]
=> [5,1,1,1]
=> 4
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> [5]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> [5]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> [7,5,1,1,1,1]
=> ? = 8
([(2,3)],4)
=> [6,6]
=> [12]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [5,1,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [8]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [5,1,1,1,1,1]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [5,1,1,1,1]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> [2,2,2,2,1]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2,2,1]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [9,4,4,4,4,4,2,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(3,4)],5)
=> [6,6,6,6]
=> [13,1,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [12,5,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,4]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [7,6,1,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [13]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [5,1,1,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [5,5]
=> 4
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [5,1,1,1,1,1,1]
=> 3
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [8,4]
=> 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [2,2,2,2,2]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [9]
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [6,1,1]
=> 3
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [9,1,1,1,1,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [8,4]
=> 4
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [5,5]
=> 4
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [12,5,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [8,4]
=> 4
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [6,1,1]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,4]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [5,5]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [7,6,1,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [9,6]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [6,1,1,1,1,1,1,1]
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [6,6]
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [6,5]
=> 5
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [5,1,1,1,1,1,1]
=> 3
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [11,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [13]
=> 2
([(1,4),(2,3)],5)
=> [6,6,6]
=> [2,2,2,2,2,2,2,2,2]
=> 3
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> 2
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [9]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [7,2,2,2,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 2
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [11,1,1,1]
=> ? ∊ {3,3,4,4,5,6,6,7,7,8,8,16}
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: 67% ●values known / values provided: 80%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 80%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
([],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 = 4 - 1
([(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 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
([(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 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7 = 8 - 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 = 2 - 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 = 3 - 1
([(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 = 4 - 1
([(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 = 1 - 1
([(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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 1
([(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 = 2 - 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 = 4 - 1
([(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 = 3 - 1
([(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 = 2 - 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 = 3 - 1
([(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 = 1 - 1
([(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 - 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7 = 8 - 1
([(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 = 2 - 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 = 3 - 1
([(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 = 4 - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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 = 3 - 1
([(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 = 4 - 1
([(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 = 2 - 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 = 3 - 1
([(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 = 2 - 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 = 3 - 1
([(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 = 4 - 1
([(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 = 4 - 1
([(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 = 4 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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 = 4 - 1
([(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 = 3 - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7 = 8 - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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 = 5 - 1
([(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 = 2 - 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 = 3 - 1
([(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 = 3 - 1
([(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 = 2 - 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 = 3 - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 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 = 2 - 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 = 2 - 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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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 = 2 - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
([(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]
=> ? ∊ {1,1,1,2,3,3,4,4,4,4,4,5,6,6,7,7,16} - 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000439
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 78%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 78%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 4 + 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> 9 = 8 + 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 3 = 2 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 5 = 4 + 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 4 = 3 + 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 5 = 4 + 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2 = 1 + 1
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(3,4)],5)
=> [6,6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5 = 4 + 1
([(1,3),(1,4),(4,2)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 5 = 4 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 5 = 4 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5 = 4 + 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 5 = 4 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 5 = 4 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 5 = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 6 = 5 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 3 = 2 + 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 6 = 5 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(1,4),(2,3),(3,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
([(0,3),(1,4),(4,2)],5)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? ∊ {1,1,1,2,2,2,2,3,3,3,3,3,3,3,4,7,7,8,8,16} + 1
Description
The position of the first down step of a Dyck path.
The following 125 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000668The least common multiple of the parts of the partition. St000288The number of ones in a binary word. St000052The number of valleys of a Dyck path not on the x-axis. St000734The last entry in the first row of a standard tableau. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000025The number of initial rises of a Dyck path. St000507The number of ascents of a standard tableau. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000676The number of odd rises of 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. St000053The number of valleys of the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. 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. St001645The pebbling number of a connected graph. St000120The number of left tunnels of a Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000264The girth of a graph, which is not a tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. St000455The second largest eigenvalue of a graph if it is integral. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001462The number of factors of a standard tableaux under concatenation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001712The number of natural descents of a standard Young tableau. St000454The largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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 $eA$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000656The number of cuts of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001637The number of (upper) dissectors of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St001597The Frobenius rank of a skew partition. St001596The number of two-by-two squares inside a skew partition. St000453The number of distinct Laplacian eigenvalues of a graph. St001093The detour number of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001341The number of edges in the center of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001691The number of kings in a graph. St001736The total number of cycles in a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000081The number of edges of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000633The size of the automorphism group of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000284The Plancherel distribution on integer partitions. 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. 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. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000299The number of nonisomorphic vertex-induced subtrees. St001649The length of a longest trail in a graph. St001869The maximum cut size of a graph. St001118The acyclic chromatic index of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000706The product of the factorials of the multiplicities of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001624The breadth of a lattice. St001330The hat guessing number of a graph.
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