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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St001212
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St001212: 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
St001212: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
([],2)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
([],3)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
([(0,2),(1,2)],3)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [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]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [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]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [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]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [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]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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]
=> 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [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
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St001208
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> [1,0]
=> [2,1] => 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 1
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ? = 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ? = 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(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]
=> [5,1,2,3,6,4] => ? = 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ? = 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 2
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001236
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> 10 => [1,2] => 1
([],2)
=> [1,1]
=> 110 => [1,1,2] => 1
([(0,1)],2)
=> [2]
=> 100 => [1,3] => 1
([],3)
=> [1,1,1]
=> 1110 => [1,1,1,2] => 1
([(1,2)],3)
=> [2,1]
=> 1010 => [1,2,2] => 1
([(0,2),(1,2)],3)
=> [3]
=> 1000 => [1,4] => 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => [1,4] => 1
([],4)
=> [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 1
([(2,3)],4)
=> [2,1,1]
=> 10110 => [1,2,1,2] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => [1,3,2] => 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> 10000 => [1,5] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => [1,1,3] => 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> 10000 => [1,5] => 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => [1,3,2] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => [1,5] => 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => [1,5] => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => [1,5] => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => [1,5] => 1
([],5)
=> [1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => ? = 1
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,4,2] => ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => [1,1,2,2] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => [1,4,2] => ? = 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => [1,2,3] => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => [1,3,1,2] => ? = 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,4,2] => ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => [1,4,2] => ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,4,2] => ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => [1,2,3] => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,4,2] => ? = 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => [1,6] => ? = 1
([(4,5)],6)
=> [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 2
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> 100100 => [1,3,3] => ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1000000 => [1,7] => ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 11000 => [1,1,4] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1000000 => [1,7] => ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1000000 => [1,7] => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1000000 => [1,7] => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 1000000 => [1,7] => ? = 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 11100 => [1,1,1,3] => 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 1000010 => [1,5,2] => ? = 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> 100100 => [1,3,3] => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> 11000 => [1,1,4] => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> 11000 => [1,1,4] => 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St001371
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> [1,0]
=> 10 => 0 = 1 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0 = 1 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0 = 1 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0 = 1 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(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]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 1 - 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0 = 1 - 1
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001570
Values
([],1)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 - 1
([],2)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 - 1
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([],3)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 - 1
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 - 1
([],4)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 - 1
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
([],5)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 - 1
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> ? = 1 - 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ?
=> ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ?
=> ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ?
=> ? = 1 - 1
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 0 = 1 - 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 0 = 1 - 1
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001730
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> [1,0]
=> 10 => 0 = 1 - 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> 1100 => 0 = 1 - 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 1010 => 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 0 = 1 - 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 101010 => 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 0 = 1 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0 = 1 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0 = 1 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 0 = 1 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0 = 1 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(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]
=> 1010101100 => ? = 1 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 1 - 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 2 - 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 2 - 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1 - 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 2 - 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 0 = 1 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 0 = 1 - 1
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001060
Values
([],1)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 + 2
([],2)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 + 2
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([],3)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 + 2
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 1 + 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 + 2
([],4)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 + 2
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 1 + 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 + 2
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 + 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
([],5)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 1 + 2
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 2 + 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> ? = 1 + 2
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 + 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 1 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ?
=> ? = 1 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> ? = 1 + 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 1 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ?
=> ? = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ?
=> ? = 1 + 2
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 + 2
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? = 2 + 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> ? = 2 + 2
([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 3 = 1 + 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,6),(2,5),(3,4)],7)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001195
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> [1,0]
=> [1,1,0,0]
=> ? = 1
([],2)
=> [1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
([(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(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]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001487
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> [[1],[]]
=> 1
([],2)
=> [1,1]
=> [[1,1],[]]
=> 1
([(0,1)],2)
=> [2]
=> [[2],[]]
=> 1
([],3)
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 1
([(0,2),(1,2)],3)
=> [2,2]
=> [[2,2],[]]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[2,2],[]]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [[3,2],[]]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[3,3,3],[]]
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[3,3,3],[]]
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [[4,2],[]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [[4,3],[]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [[4,4],[]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[5],[]]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2
([(2,5),(3,4),(4,5)],6)
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,1]
=> [[3,3,3,1],[]]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
([],1)
=> [1]
=> [[1],[]]
=> 1
([],2)
=> [1,1]
=> [[1,1],[]]
=> 1
([(0,1)],2)
=> [2]
=> [[2],[]]
=> 1
([],3)
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[2,1],[]]
=> 1
([(0,2),(1,2)],3)
=> [2,2]
=> [[2,2],[]]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[3],[]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
([(1,3),(2,3)],4)
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[2,2],[]]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[3,1],[]]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,2]
=> [[3,2],[]]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [[4],[]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1
([(2,4),(3,4)],5)
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2
([(1,4),(2,4),(3,4)],5)
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[2,2,1],[]]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
([(0,1),(2,4),(3,4)],5)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[3,1,1],[]]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> [[3,3,1],[]]
=> ? = 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[3,3,3],[]]
=> ? = 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[3,2],[]]
=> 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,2,2]
=> [[3,2,2],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[3,3],[]]
=> ? = 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> [[3,3,3],[]]
=> ? = 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> [[3,3,2],[]]
=> ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[4,1],[]]
=> 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> [[4,2],[]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,3]
=> [[4,3],[]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> [[4,4],[]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[5],[]]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 1
([(3,5),(4,5)],6)
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 2
([(2,5),(3,5),(4,5)],6)
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 2
([(2,5),(3,4),(4,5)],6)
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 2
([(1,2),(3,5),(4,5)],6)
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? = 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,1,1]
=> [[3,3,1,1],[]]
=> ? = 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,1]
=> [[3,2,2,1],[]]
=> ? = 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,1]
=> [[3,3,2,1],[]]
=> ? = 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,2,2,2]
=> [[3,2,2,2],[]]
=> ? = 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,1]
=> [[3,3,3,1],[]]
=> ? = 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,2,2]
=> [[3,3,2,2],[]]
=> ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3,3,2]
=> [[3,3,3,2],[]]
=> ? = 2
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
Description
The number of connected components of a skew partition.
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