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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001140
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Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001140: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001140: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> 0
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 0
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
Description
Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001875
Values
([],1)
=> ([],1)
=> ([],1)
=> ? = 0 + 3
([],2)
=> ([],1)
=> ([],1)
=> ? = 0 + 3
([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 0 + 3
([],3)
=> ([],1)
=> ([],1)
=> ? = 0 + 3
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(0,2),(1,2)],3)
=> ([],2)
=> ([],1)
=> ? = 0 + 3
([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 0 + 3
([],4)
=> ([],1)
=> ([],1)
=> ? = 0 + 3
([(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(0,3),(1,3),(2,3)],4)
=> ([],2)
=> ([],1)
=> ? = 0 + 3
([(0,3),(1,2)],4)
=> ([],4)
=> ([],1)
=> ? = 0 + 3
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],2)
=> ([],1)
=> ? = 0 + 3
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],3)
=> ([],1)
=> ? = 0 + 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> ? = 1 + 3
([],5)
=> ([],1)
=> ([],1)
=> ? = 1 + 3
([(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 2 + 3
([(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ([],1)
=> ? = 0 + 3
([(1,4),(2,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 3
([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(2,4),(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 0 + 3
([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 + 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? = 2 + 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> ? = 0 + 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 0 + 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(2,3),(2,4)],5)
=> ([],1)
=> ? = 0 + 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> ? = 0 + 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> ? = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([],5)
=> ([],1)
=> ? = 0 + 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 0 + 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 + 3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],1)
=> ? = 0 + 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([],3)
=> ([],1)
=> ? = 0 + 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],4)
=> ([],1)
=> ? = 0 + 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> ? = 2 + 3
([],6)
=> ([],1)
=> ([],1)
=> ? = 3 + 3
([(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 3 + 3
([(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 1 + 3
([(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 0 + 3
([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(2,4),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
([(0,5),(1,3),(1,4),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
Description
The number of simple modules with projective dimension at most 1.
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