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Your data matches 15 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([],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
([(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
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
([(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
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
([],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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
([(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
Description
The position of the first down step of a Dyck path.
Matching statistic: St000025
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 67%●distinct values known / distinct values provided: 62%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 67%●distinct values known / distinct values provided: 62%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 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]
=> 4 = 5 - 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]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 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]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 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 - 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 - 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]
=> 3 = 4 - 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]
=> 4 = 5 - 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
([(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]
=> 2 = 3 - 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]
=> 2 = 3 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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]
=> 2 = 3 - 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]
=> 4 = 5 - 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]
=> 3 = 4 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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]
=> 1 = 2 - 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
([(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 - 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 - 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]
=> 4 = 5 - 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]
=> 4 = 5 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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 - 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]
=> 4 = 5 - 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]
=> 4 = 5 - 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 - 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]
=> 4 = 5 - 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]
=> 3 = 4 - 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]
=> 4 = 5 - 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]
=> 4 = 5 - 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]
=> 4 = 5 - 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]
=> 5 = 6 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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 - 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]
=> 2 = 3 - 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]
=> 5 = 6 - 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 - 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]
=> 3 = 4 - 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
([(0,4),(1,2),(1,3)],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]
=> 4 = 5 - 1
([(0,4),(1,2),(1,3),(1,4)],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]
=> 3 = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],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]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],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]
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],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]
=> 5 = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],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
([(0,3),(0,4),(1,2),(2,3),(2,4)],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 - 1
([(1,4),(3,2),(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
([(0,3),(3,4),(4,1),(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]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],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
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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]
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],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]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(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]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],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]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 4 = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [7,2,2,2,2]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> [5,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> [8,6]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3 - 1
([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,5),(3,5)],6)
=> [8,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> [8,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
([(1,5),(2,3),(3,5),(5,4)],6)
=> [8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3 - 1
([(1,5),(4,3),(5,2),(5,4)],6)
=> [8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 3 - 1
([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [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 - 1
([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> [8,6]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 3 - 1
([(1,5),(3,4),(4,2),(5,3)],6)
=> [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 - 1
([(0,5),(1,4),(4,2),(5,3)],6)
=> [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 - 1
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [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
([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 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]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
([(0,3),(0,4),(3,5),(3,6),(4,2),(4,5),(4,6),(6,1)],7)
=> [9,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 3 - 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 5 - 1
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 9 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> 3 = 4 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 2 = 3 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> 3 = 4 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 3 = 4 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1 = 2 - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
=> [[1,3,5,7,9,15],[2,4,6,8,10,16],[11,17],[12,18],[13,19],[14,20]]
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 4 = 5 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 3 = 4 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> [[1,5,9,13],[2,6,10,14],[3,7,11,15],[4,8,12,16]]
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 5 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 4 = 5 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
=> [[1,3,5,7,9,15],[2,4,6,8,10,16],[11,17],[12,18],[13,19],[14,20]]
=> ? = 7 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 5 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 4 = 5 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 3 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11],[12],[13],[14]]
=> ? = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 3 = 4 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 5 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> 2 = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 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]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [[1,2,11,12,13,14,15],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 3 = 4 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 5 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 5 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> ? = 4 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [[1,2,3,4,5,6,13,14],[7,8,9,10,11,12]]
=> ?
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 5 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5,4,4,4]
=> [[1,2,3,4,17],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ?
=> ? = 5 - 1
([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 5 - 1
([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ?
=> ? = 4 - 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 4 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 4 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [7,2,2,2,2]
=> [[1,2,11,12,13,14,15],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000307
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1 = 2 - 1
([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 2 - 1
([],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(1,2)],3)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 8 = 9 - 1
([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 2 = 3 - 1
([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 3 = 4 - 1
([(0,1),(0,2),(0,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 3 - 1
([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 3 = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 4 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1 = 2 - 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 2 = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 4 = 5 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ? = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 3 = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 4 - 1
([(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7)
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7)
=> ? = 5 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 4 = 5 - 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? = 7 - 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 5 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 4 = 5 - 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
=> ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ? = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 4 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ? = 5 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8)
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
=> ([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(7,2),(7,3)],8)
=> ? = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1 = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> ? = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
=> ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ([(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2)],8)
=> ? = 6 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 3 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 4 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7)
=> ? = 5 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7)
=> ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,6),(6,2),(6,3)],8)
=> ? = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 6 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ? = 5 - 1
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,1),(6,2),(6,3)],8)
=> ? = 6 - 1
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 6 - 1
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 5 - 1
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
=> ([(0,2),(0,3),(0,4),(1,6),(2,7),(3,7),(4,5),(5,6),(7,1),(7,5)],8)
=> ? = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,7),(4,7),(5,7),(7,1),(7,2)],8)
=> ? = 6 - 1
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,2),(7,1),(7,5)],8)
=> ? = 3 - 1
([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ? = 4 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7)
=> ? = 5 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 3 = 4 - 1
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
=> ? = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 3 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 2 = 3 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2 = 3 - 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 $P \setminus I$.
Matching statistic: St000329
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> 0 = 2 - 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 3 - 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> 0 = 2 - 2
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 5 - 2
([(1,2)],3)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([],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]
=> ? = 9 - 2
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 5 - 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 5 - 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2 = 4 - 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 4 - 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 3 - 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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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 = 3 - 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 4 - 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 = 3 - 2
([(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]
=> 2 = 4 - 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 5 - 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 4 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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]
=> ? = 6 - 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 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]
=> ? = 4 - 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]
=> ? = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 6 - 2
([(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]
=> ? = 3 - 2
([(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]
=> 2 = 4 - 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]
=> ? = 2 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 4 - 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 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]
=> ? = 5 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 6 - 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 2
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 2 - 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 4 - 2
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 2
([(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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 6 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(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]
=> ? = 6 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4 - 2
([(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]
=> ? = 5 - 2
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 6 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1 = 3 - 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000015
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 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]
=> 4 = 5 - 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]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 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]
=> ? = 9 - 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]
=> ? = 3 - 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]
=> ? = 4 - 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]
=> 4 = 5 - 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]
=> ? = 2 - 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]
=> 2 = 3 - 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]
=> 2 = 3 - 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]
=> 2 = 3 - 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]
=> ? = 4 - 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]
=> 2 = 3 - 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]
=> 4 = 5 - 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]
=> 3 = 4 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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]
=> 1 = 2 - 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]
=> ? = 2 - 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]
=> ? = 7 - 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]
=> ? = 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]
=> ? = 5 - 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]
=> ? = 5 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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]
=> 2 = 3 - 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]
=> 3 = 4 - 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]
=> ? = 5 - 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]
=> ? = 5 - 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]
=> ? = 5 - 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]
=> ? = 7 - 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]
=> ? = 5 - 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]
=> 3 = 4 - 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]
=> ? = 5 - 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]
=> ? = 5 - 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]
=> ? = 5 - 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]
=> ? = 6 - 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]
=> 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]
=> ? = 4 - 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]
=> ? = 3 - 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]
=> 2 = 3 - 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]
=> ? = 6 - 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]
=> ? = 3 - 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]
=> 3 = 4 - 1
([(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]
=> ? = 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]
=> ? = 5 - 1
([(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]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],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]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],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]
=> ? = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],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]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 3 - 1
([(1,4),(3,2),(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]
=> ? = 2 - 1
([(0,3),(3,4),(4,1),(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]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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]
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],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]
=> 3 = 4 - 1
([(0,4),(1,2),(2,3),(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]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],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]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
([(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]
=> ?
=> ? = 5 - 1
([(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]
=> ? = 5 - 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]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
([(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]
=> ? = 5 - 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]
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 1
([(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]
=> ?
=> ? = 5 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
([(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]
=> ? = 5 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 3 - 1
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 3 - 1
Description
The number of peaks of a Dyck path.
Matching statistic: St001462
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
([(0,1)],2)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 4 = 5 - 1
([(1,2)],3)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([(0,2),(2,1)],3)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2 = 3 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 9 - 1
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
=> ? = 3 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> ? = 4 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> ? = 5 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 2 = 3 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> ? = 4 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> ? = 5 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 4 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 3 = 4 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 2 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1 = 2 - 1
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
=> [[1,3,5,7,9,15],[2,4,6,8,10,16],[11,17],[12,18],[13,19],[14,20]]
=> ? = 7 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> ? = 5 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 5 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> ? = 4 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 3 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 3 = 4 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> [[1,5,9,13],[2,6,10,14],[3,7,11,15],[4,8,12,16]]
=> ? = 5 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 5 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> ? = 5 - 1
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
=> [[1,3,5,7,9,15],[2,4,6,8,10,16],[11,17],[12,18],[13,19],[14,20]]
=> ? = 7 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 5 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> ? = 5 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 6 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 4 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 3 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11],[12],[13],[14]]
=> ? = 6 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 3 = 4 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 5 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 4 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 6 - 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> ? = 3 - 1
([(1,4),(3,2),(4,3)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> ? = 4 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> ? = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2 = 3 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 5 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 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]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 4 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 5 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [[1,2,11,12,13,14,15],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 6 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> ? = 4 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> ? = 3 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> ? = 4 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 5 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2 = 3 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2 = 3 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2 = 3 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1 = 2 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2 = 3 - 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 1 = 2 - 1
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St000331
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 2 - 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
([],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 = 5 - 2
([(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 = 2 - 2
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
([],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]
=> ? = 9 - 2
([(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]
=> ? = 3 - 2
([(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]
=> ? = 4 - 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 = 5 - 2
([(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]
=> ? = 2 - 2
([(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 = 3 - 2
([(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 = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 4 - 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 = 3 - 2
([(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 = 5 - 2
([(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 = 4 - 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 = 3 - 2
([(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 = 4 - 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 = 2 - 2
([(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]
=> ? = 2 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 3 - 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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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 = 3 - 2
([(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 = 4 - 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 = 3 - 2
([(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 = 4 - 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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 5 - 2
([(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 = 4 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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]
=> ? = 6 - 2
([(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 = 3 - 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]
=> ? = 4 - 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]
=> ? = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 6 - 2
([(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]
=> ? = 3 - 2
([(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 = 4 - 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]
=> ? = 2 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 4 - 2
([(0,2),(0,4),(3,1),(4,3)],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 = 3 - 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 5 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],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]
=> ? = 6 - 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],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]
=> ? = 2 - 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 3 - 2
([(1,4),(3,2),(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]
=> ? = 2 - 2
([(0,3),(3,4),(4,1),(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 = 3 - 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
([(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 2 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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 = 2 - 2
([(0,3),(1,2),(2,4),(3,4)],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 = 4 - 2
([(0,4),(1,2),(2,3),(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 = 3 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],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 = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
([(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]
=> ?
=> ? = 5 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(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]
=> ?
=> ? = 6 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
([(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]
=> ?
=> ? = 5 - 2
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
Description
The number of upper interactions of a Dyck path.
An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St001169
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 2 - 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
([],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 = 5 - 2
([(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 = 2 - 2
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
([],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]
=> ? = 9 - 2
([(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]
=> ? = 3 - 2
([(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]
=> ? = 4 - 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 = 5 - 2
([(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]
=> ? = 2 - 2
([(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 = 3 - 2
([(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 = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 4 - 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 = 3 - 2
([(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 = 5 - 2
([(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 = 4 - 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 = 3 - 2
([(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 = 4 - 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 = 2 - 2
([(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]
=> ? = 2 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 3 - 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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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 = 3 - 2
([(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 = 4 - 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 = 3 - 2
([(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 = 4 - 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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 5 - 2
([(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 = 4 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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]
=> ? = 6 - 2
([(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 = 3 - 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]
=> ? = 4 - 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]
=> ? = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 6 - 2
([(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]
=> ? = 3 - 2
([(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 = 4 - 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]
=> ? = 2 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 4 - 2
([(0,2),(0,4),(3,1),(4,3)],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 = 3 - 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 5 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],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]
=> ? = 6 - 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],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]
=> ? = 2 - 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 3 - 2
([(1,4),(3,2),(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]
=> ? = 2 - 2
([(0,3),(3,4),(4,1),(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 = 3 - 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
([(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 2 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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 = 2 - 2
([(0,3),(1,2),(2,4),(3,4)],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 = 4 - 2
([(0,4),(1,2),(2,3),(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 = 3 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],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 = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
([(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]
=> ?
=> ? = 5 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(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]
=> ?
=> ? = 6 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
([(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]
=> ?
=> ? = 5 - 2
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001227
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
([],1)
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 2 - 2
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 3 - 2
([(0,1)],2)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
([],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 = 5 - 2
([(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 = 2 - 2
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
([(0,2),(2,1)],3)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 3 - 2
([],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]
=> ? = 9 - 2
([(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]
=> ? = 3 - 2
([(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]
=> ? = 4 - 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 = 5 - 2
([(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]
=> ? = 2 - 2
([(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 = 3 - 2
([(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 = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 4 - 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 = 3 - 2
([(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 = 5 - 2
([(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 = 4 - 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 = 3 - 2
([(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 = 4 - 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 = 2 - 2
([(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]
=> ? = 2 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 3 - 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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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 = 3 - 2
([(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 = 4 - 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 = 3 - 2
([(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 = 4 - 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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 7 - 2
([(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]
=> ? = 5 - 2
([(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 = 4 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 5 - 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]
=> ? = 5 - 2
([(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]
=> ? = 6 - 2
([(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 = 3 - 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]
=> ? = 4 - 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]
=> ? = 3 - 2
([(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 = 3 - 2
([(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]
=> ? = 6 - 2
([(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]
=> ? = 3 - 2
([(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 = 4 - 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]
=> ? = 2 - 2
([(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]
=> ? = 5 - 2
([(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]
=> ? = 4 - 2
([(0,2),(0,4),(3,1),(4,3)],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 = 3 - 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 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]
=> ? = 5 - 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],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]
=> ? = 6 - 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],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]
=> ? = 2 - 2
([(0,3),(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 3 - 2
([(1,4),(3,2),(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]
=> ? = 2 - 2
([(0,3),(3,4),(4,1),(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 = 3 - 2
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 - 2
([(0,4),(1,2),(2,3),(2,4)],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]
=> ? = 2 - 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [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 = 2 - 2
([(0,3),(1,2),(2,4),(3,4)],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 = 4 - 2
([(0,4),(1,2),(2,3),(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 = 3 - 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],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 = 3 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
([(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]
=> ?
=> ? = 5 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(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]
=> ?
=> ? = 6 - 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 3 - 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 4 - 2
([(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]
=> ?
=> ? = 5 - 2
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 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]
=> ? = 5 - 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 - 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 3 - 2
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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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