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Your data matches 80 different statistics following compositions of up to 3 maps.
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Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 1
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 1
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 2
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 2 = 1 + 1
([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 2 = 1 + 1
([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> 2 = 1 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 2 = 1 + 1
([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 2 = 1 + 1
([(4,5)],6)
=> [1]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 2 = 1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 2 = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 3 = 2 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3 = 2 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 3 = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3 = 2 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(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
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The number of valleys of the Dyck path.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000291: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 10 => 1
([(1,2)],3)
=> [1]
=> 10 => 1
([(0,2),(1,2)],3)
=> [1,1]
=> 110 => 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(2,3)],4)
=> [1]
=> 10 => 1
([(1,3),(2,3)],4)
=> [1,1]
=> 110 => 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 1110 => 1
([(0,3),(1,2)],4)
=> [1,1]
=> 110 => 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 1110 => 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1000 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1
([(3,4)],5)
=> [1]
=> 10 => 1
([(2,4),(3,4)],5)
=> [1,1]
=> 110 => 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 11110 => 1
([(1,4),(2,3)],5)
=> [1,1]
=> 110 => 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1110 => 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1000 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 11110 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 10000 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 11110 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => 1
([(4,5)],6)
=> [1]
=> 10 => 1
([(3,5),(4,5)],6)
=> [1,1]
=> 110 => 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 1
([(2,5),(3,4)],6)
=> [1,1]
=> 110 => 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 1110 => 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1000 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 10010 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 100110 => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 10000 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 11110 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> 100010 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 100110 => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 1000110 => 2
Description
The number of descents of a binary word.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(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
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The bounce count of a Dyck path. For a Dyck path D of length 2n, this is the number of points (i,i) for 1i<n that are touching points of the [[Mp00099|bounce path]] of D.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000390: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 10 => 1
([(1,2)],3)
=> [1]
=> 10 => 1
([(0,2),(1,2)],3)
=> [1,1]
=> 110 => 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 1
([(2,3)],4)
=> [1]
=> 10 => 1
([(1,3),(2,3)],4)
=> [1,1]
=> 110 => 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 1110 => 1
([(0,3),(1,2)],4)
=> [1,1]
=> 110 => 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 1110 => 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1000 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1
([(3,4)],5)
=> [1]
=> 10 => 1
([(2,4),(3,4)],5)
=> [1,1]
=> 110 => 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 11110 => 1
([(1,4),(2,3)],5)
=> [1,1]
=> 110 => 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1110 => 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1000 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 11110 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 10000 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 11110 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => 1
([(4,5)],6)
=> [1]
=> 10 => 1
([(3,5),(4,5)],6)
=> [1,1]
=> 110 => 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 1
([(2,5),(3,4)],6)
=> [1,1]
=> 110 => 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 1110 => 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1000 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 10010 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 100110 => 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> 10000 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 11110 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> 100010 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 100110 => 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 1000110 => 2
Description
The number of runs of ones in a binary word.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(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
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(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
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(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
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
St001199: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(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
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
Description
The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
The following 70 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000015The number of peaks of a Dyck path. St000292The number of ascents of a binary word. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn1] such that n=c0<ci for all i>0 a Dyck path as follows: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000013The height of a Dyck path. St000386The number of factors DDU in a Dyck path. St000443The number of long tunnels of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000955Number of times one has Exti(D(A),A)>0 for i>0 for the corresponding LNakayama algebra. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001487The number of inner corners of a skew partition. St000068The number of minimal elements in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001720The minimal length of a chain of small intervals in a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000181The number of connected components of the Hasse diagram for the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000258The burning number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000264The girth of a graph, which is not a tree.