Processing math: 88%

Your data matches 25 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00318: Graphs dual on componentsGraphs
St000081: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],2)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],3)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],4)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
([],5)
=> ([],5)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
([(0,3),(0,4),(1,2),(1,3),(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)
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 8
Description
The number of edges of a graph.
Matching statistic: St000459
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> 0
([],2)
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([],4)
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([],5)
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [3]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
Description
The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Mp00318: Graphs dual on componentsGraphs
Mp00203: Graphs coneGraphs
St001311: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> ([(0,1)],2)
=> 0
([],2)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],3)
=> ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([],5)
=> ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(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)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(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),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(0,3),(0,4),(1,2),(1,3),(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
=> 8
Description
The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as c+mn, where c is the number of connected components, m is the number of edges and n is the number of vertices.
Mp00259: Graphs vertex additionGraphs
St001341: Graphs ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],2)
=> 0
([],2)
=> ([],3)
=> 0
([(0,1)],2)
=> ([(1,2)],3)
=> 1
([],3)
=> ([],4)
=> 0
([(1,2)],3)
=> ([(2,3)],4)
=> 1
([(0,2),(1,2)],3)
=> ([(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(1,2),(1,3),(2,3)],4)
=> 3
([],4)
=> ([],5)
=> 0
([(2,3)],4)
=> ([(3,4)],5)
=> 1
([(1,3),(2,3)],4)
=> ([(2,4),(3,4)],5)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2)],4)
=> ([(1,4),(2,3)],5)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(1,4),(2,3),(3,4)],5)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(2,3),(2,4),(3,4)],5)
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
([],5)
=> ([],6)
=> 0
([(3,4)],5)
=> ([(4,5)],6)
=> 1
([(2,4),(3,4)],5)
=> ([(3,5),(4,5)],6)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(2,5),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(1,4),(2,3)],5)
=> ([(2,5),(3,4)],6)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(2,5),(3,4),(4,5)],6)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(1,2),(3,5),(4,5)],6)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 8
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 9
Description
The number of edges in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.
Mp00276: Graphs to edge-partition of biconnected componentsInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001034: Dyck paths ⟶ ℤResult quality: 92% values known / values provided: 99%distinct values known / distinct values provided: 92%
Values
([],1)
=> []
=> []
=> []
=> 0
([],2)
=> []
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1]
=> [1,0]
=> 1
([],3)
=> []
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1]
=> [1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([],4)
=> []
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1]
=> [1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
([],5)
=> []
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1]
=> [1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 8
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
Description
The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
St000228: Integer partitions ⟶ ℤResult quality: 85% values known / values provided: 98%distinct values known / distinct values provided: 85%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 6
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 8
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 11
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [11]
=> ? = 11
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> ? = 12
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000460
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000460: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 97%distinct values known / distinct values provided: 92%
Values
([],1)
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> []
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([],4)
=> []
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([],5)
=> []
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [3]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 10
([],6)
=> []
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 4
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000870: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 97%distinct values known / distinct values provided: 92%
Values
([],1)
=> []
=> []
=> ? = 0
([],2)
=> []
=> []
=> ? = 0
([(0,1)],2)
=> [1]
=> [1]
=> 1
([],3)
=> []
=> []
=> ? = 0
([(1,2)],3)
=> [1]
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 3
([],4)
=> []
=> []
=> ? = 0
([(2,3)],4)
=> [1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([],5)
=> []
=> []
=> ? = 0
([(3,4)],5)
=> [1]
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> [3]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 8
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 10
([],6)
=> []
=> []
=> ? = 0
([(4,5)],6)
=> [1]
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 4
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells (i,i) of a partition.
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000293: Binary words ⟶ ℤResult quality: 77% values known / values provided: 95%distinct values known / distinct values provided: 77%
Values
([],1)
=> []
=> => ? = 0
([],2)
=> []
=> => ? = 0
([(0,1)],2)
=> [1]
=> 10 => 1
([],3)
=> []
=> => ? = 0
([(1,2)],3)
=> [1]
=> 10 => 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 3
([],4)
=> []
=> => ? = 0
([(2,3)],4)
=> [1]
=> 10 => 1
([(1,3),(2,3)],4)
=> [2]
=> 100 => 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1000 => 3
([(0,3),(1,2)],4)
=> [1,1]
=> 110 => 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1000 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 100000 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1000000 => 6
([],5)
=> []
=> => ? = 0
([(3,4)],5)
=> [1]
=> 10 => 1
([(2,4),(3,4)],5)
=> [2]
=> 100 => 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1000 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 10000 => 4
([(1,4),(2,3)],5)
=> [1,1]
=> 110 => 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1000 => 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 1010 => 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1000 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 10000 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 10000 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 10000 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1000000 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 10000000 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 10000 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 100000 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 10000000 => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 10000000 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 100000000 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 10000000 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 100000000 => 8
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 10000000000 => 10
([],6)
=> []
=> => ? = 0
([(4,5)],6)
=> [1]
=> 10 => 1
([(3,5),(4,5)],6)
=> [2]
=> 100 => 2
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 1000 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 10000 => 4
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> 100000000000 => ? = 11
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [11]
=> 100000000000 => ? = 11
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> 1000000000000 => ? = 12
Description
The number of inversions of a binary word.
Mp00275: Graphs to edge-partition of connected componentsInteger partitions
Mp00095: Integer partitions to binary wordBinary words
Mp00316: Binary words inverse Foata bijectionBinary words
St000290: Binary words ⟶ ℤResult quality: 77% values known / values provided: 95%distinct values known / distinct values provided: 77%
Values
([],1)
=> []
=> => ? => ? = 0
([],2)
=> []
=> => ? => ? = 0
([(0,1)],2)
=> [1]
=> 10 => 10 => 1
([],3)
=> []
=> => ? => ? = 0
([(1,2)],3)
=> [1]
=> 10 => 10 => 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 010 => 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 0010 => 3
([],4)
=> []
=> => ? => ? = 0
([(2,3)],4)
=> [1]
=> 10 => 10 => 1
([(1,3),(2,3)],4)
=> [2]
=> 100 => 010 => 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 1000 => 0010 => 3
([(0,3),(1,2)],4)
=> [1,1]
=> 110 => 110 => 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1000 => 0010 => 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1000 => 0010 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 10000 => 00010 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 00010 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 100000 => 000010 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [6]
=> 1000000 => 0000010 => 6
([],5)
=> []
=> => ? => ? = 0
([(3,4)],5)
=> [1]
=> 10 => 10 => 1
([(2,4),(3,4)],5)
=> [2]
=> 100 => 010 => 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(1,4),(2,3)],5)
=> [1,1]
=> 110 => 110 => 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 1010 => 0110 => 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1000 => 0010 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 000010 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 000010 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 000010 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 000010 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 10000000 => 00000010 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 10000 => 00010 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 00110 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> 100000 => 000010 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 100000 => 000010 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 10000000 => 00000010 => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 1000000 => 0000010 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 10000000 => 00000010 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 100000000 => 000000010 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [7]
=> 10000000 => 00000010 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [8]
=> 100000000 => 000000010 => 8
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [10]
=> 10000000000 => 00000000010 => 10
([],6)
=> []
=> => ? => ? = 0
([(4,5)],6)
=> [1]
=> 10 => 10 => 1
([(3,5),(4,5)],6)
=> [2]
=> 100 => 010 => 2
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 1000 => 0010 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 10000 => 00010 => 4
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> 100000000000 => ? => ? = 11
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> [11]
=> 100000000000 => ? => ? = 11
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [12]
=> 1000000000000 => ? => ? = 12
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length n with a zeros, the generating function for the major index is the q-binomial coefficient \binom{n}{a}_q.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000395The sum of the heights of the peaks of a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000171The degree of the graph. St001746The coalition number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001725The harmonious chromatic number of a graph. St000456The monochromatic index of a connected graph. St001622The number of join-irreducible elements of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset.