searching the database
Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> ([],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
([],5)
=> ([],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
([],6)
=> ([],6)
=> ([],6)
=> ([(0,1),(0,2),(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)
=> 3
([(4,5)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> ([(0,2),(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)
=> 3
([(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(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)
=> 3
([(2,5),(3,4),(4,5)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(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)
=> 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 4
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
([],7)
=> ([],7)
=> ([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(5,6)],7)
=> ([(5,6)],7)
=> ([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(3,6),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(3,6),(4,5),(5,6)],7)
=> ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001137
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001137: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00030: Dyck paths āzeta mapā¶ Dyck paths
St001137: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 3 - 3
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 3 - 3
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 3 - 3
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 4 - 3
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 3 - 3
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 3 - 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 4 - 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 4 - 3
([],6)
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0 = 3 - 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 0 = 3 - 3
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
([(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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
([(2,3),(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]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 4 - 3
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0 = 3 - 3
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 0 = 3 - 3
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 3 - 3
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 4 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 0 = 3 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0 = 3 - 3
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St001939
Mp00152: Graphs āLaplacian multiplicitiesā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
St001939: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
St001939: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],3)
=> [3] => ([],3)
=> [1,1,1]
=> 0 = 3 - 3
([],4)
=> [4] => ([],4)
=> [1,1,1,1]
=> 0 = 3 - 3
([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 0 = 3 - 3
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 4 - 3
([],5)
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0 = 3 - 3
([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 0 = 3 - 3
([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 0 = 3 - 3
([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 4 - 3
([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 4 - 3
([],6)
=> [6] => ([],6)
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
([(4,5)],6)
=> [1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> 0 = 3 - 3
([(3,5),(4,5)],6)
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(2,5),(3,4)],6)
=> [2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
=> [2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([],7)
=> [7] => ([],7)
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
([(5,6)],7)
=> [1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> 0 = 3 - 3
([(4,6),(5,6)],7)
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(3,6),(4,5)],7)
=> [2,5] => ([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
=> [2,5] => ([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> 1 = 4 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
Description
The number of parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001940
Mp00152: Graphs āLaplacian multiplicitiesā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
St001940: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00184: Integer compositions āto threshold graphā¶ Graphs
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
St001940: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],3)
=> [3] => ([],3)
=> [1,1,1]
=> 0 = 3 - 3
([],4)
=> [4] => ([],4)
=> [1,1,1,1]
=> 0 = 3 - 3
([(2,3)],4)
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 0 = 3 - 3
([(0,3),(1,2)],4)
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 1 = 4 - 3
([],5)
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0 = 3 - 3
([(3,4)],5)
=> [1,4] => ([(3,4)],5)
=> [2,1,1,1]
=> 0 = 3 - 3
([(2,4),(3,4)],5)
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 0 = 3 - 3
([(1,4),(2,3)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 4 - 3
([(2,3),(2,4),(3,4)],5)
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1 = 4 - 3
([],6)
=> [6] => ([],6)
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
([(4,5)],6)
=> [1,5] => ([(4,5)],6)
=> [2,1,1,1,1]
=> 0 = 3 - 3
([(3,5),(4,5)],6)
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(2,5),(3,4)],6)
=> [2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
=> [2,4] => ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1 = 4 - 3
([],7)
=> [7] => ([],7)
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
([(5,6)],7)
=> [1,6] => ([(5,6)],7)
=> [2,1,1,1,1,1]
=> 0 = 3 - 3
([(4,6),(5,6)],7)
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(3,6),(4,5)],7)
=> [2,5] => ([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
=> [2,5] => ([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6,1]
=> 1 = 4 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> 0 = 3 - 3
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St000929
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs āto partition of connected componentsā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St000929: Integer partitions ā¶ ā¤Result quality: 99% āvalues known / values provided: 99%ādistinct values known / distinct values provided: 100%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00044: Integer partitions āconjugateā¶ Integer partitions
St000929: Integer partitions ā¶ ā¤Result quality: 99% āvalues known / values provided: 99%ādistinct values known / distinct values provided: 100%
Values
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 3 - 3
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0 = 3 - 3
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 3 - 3
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1 = 4 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1 = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([],7)
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0 = 3 - 3
([(5,6)],7)
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0 = 3 - 3
([(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(3,6),(4,5)],7)
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> [5,2]
=> [2]
=> [1,1]
=> 1 = 4 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1]
=> [1,1]
=> [2]
=> 0 = 3 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 0 = 3 - 3
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> [7]
=> []
=> []
=> ? = 3 - 3
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001827
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([],3)
=> ([],3)
=> ([],3)
=> 0 = 3 - 3
([],4)
=> ([],4)
=> ([],4)
=> ([],4)
=> 0 = 3 - 3
([(2,3)],4)
=> ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> 0 = 3 - 3
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 1 = 4 - 3
([],5)
=> ([],5)
=> ([],5)
=> ([],5)
=> 0 = 3 - 3
([(3,4)],5)
=> ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> 0 = 3 - 3
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 3 - 3
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> 1 = 4 - 3
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> 1 = 4 - 3
([],6)
=> ([],6)
=> ([],6)
=> ([],6)
=> 0 = 3 - 3
([(4,5)],6)
=> ([(4,5)],6)
=> ([(4,5)],6)
=> ([],5)
=> 0 = 3 - 3
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],4)
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([],4)
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,4),(2,4),(3,4)],5)
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,4),(3,4)],5)
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,4),(3,4)],6)
=> 1 = 4 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([],3)
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 1 = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,3),(2,3)],4)
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> 1 = 4 - 3
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(1,3),(2,3)],4)
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> 1 = 4 - 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],3)
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([],2)
=> 1 = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],2)
=> 1 = 4 - 3
([],7)
=> ([],7)
=> ([],7)
=> ([],7)
=> ? = 3 - 3
([(5,6)],7)
=> ([(5,6)],7)
=> ([(5,6)],7)
=> ([],6)
=> 0 = 3 - 3
([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ([(4,6),(5,6)],7)
=> ? = 3 - 3
([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> ? = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 3 - 3
([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([(3,6),(4,5)],7)
=> ([],5)
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ([(3,7),(4,6),(5,6),(5,7)],8)
=> ? = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([(2,3),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([],5)
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,5),(4,5)],6)
=> 0 = 3 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 4 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],4)
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 3 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 0 = 3 - 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? = 4 - 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],4)
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 3 - 3
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,4),(3,4)],5)
=> 0 = 3 - 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> 0 = 3 - 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> 0 = 3 - 3
([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([(1,6),(2,5),(3,4)],7)
=> ([],4)
=> 0 = 3 - 3
([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([],3)
=> 0 = 3 - 3
([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(1,2),(3,6),(4,5),(5,6)],7)
=> ([(2,6),(3,5),(4,5),(4,6)],7)
=> ? = 3 - 3
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ([(0,6),(1,5),(2,4),(3,4),(5,7),(6,7)],8)
=> ? = 4 - 3
Description
The number of two-component spanning forests of a graph.
A '''spanning subgraph''' is a subgraph which contains all vertices of the ambient graph. A '''forest''' is a graph which contains no cycles, and has any number of connected components. A '''two-component spanning forest''' is a spanning subgraph which contains no cycles and has two connected components.
Matching statistic: St001642
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1 = 3 - 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1 = 3 - 2
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> 1 = 3 - 2
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(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)
=> 1 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([],6)
=> ([(0,1),(0,2),(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)
=> ([],1)
=> 1 = 3 - 2
([(4,5)],6)
=> ([(0,2),(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)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(2,5),(3,4)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(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),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> 2 = 4 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(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)
=> 1 = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 4 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 4 - 2
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> 1 = 3 - 2
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 4 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 4 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10)],12)
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8)],10)
=> ? = 4 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 2
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 2
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 4 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 2
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 - 2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 4 - 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 4 - 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,11),(2,3),(2,5),(2,8),(2,10),(3,4),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,10),(6,7),(6,11),(7,8),(9,10),(9,11),(10,11)],12)
=> ? = 4 - 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 4 - 2
Description
The Prague dimension of a graph.
This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product.
Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.
Matching statistic: St001029
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(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)
=> 2 = 4 - 2
([(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 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([],6)
=> ([(0,1),(0,2),(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)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(4,5)],6)
=> ([(0,2),(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)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,5),(3,4)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(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),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> ([(2,5),(3,4)],6)
=> 2 = 4 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(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)
=> ([],6)
=> 1 = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(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)],6)
=> 2 = 4 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 2
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? = 3 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 4 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10)],12)
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8)],10)
=> ?
=> ? = 4 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 2
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,4),(3,6),(4,5),(6,7)],8)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([],9)
=> ? = 3 - 2
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 4 - 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? = 3 - 2
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,11),(2,3),(2,5),(2,8),(2,10),(3,4),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,10),(6,7),(6,11),(7,8),(9,10),(9,11),(10,11)],12)
=> ?
=> ? = 4 - 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ?
=> ? = 4 - 2
Description
The size of the core of a graph.
The core of the graph $G$ is the smallest graph $C$ such that there is a graph homomorphism from $G$ to $C$ and a graph homomorphism from $C$ to $G$.
Matching statistic: St001494
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(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)
=> 2 = 4 - 2
([(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 = 3 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([],6)
=> ([(0,1),(0,2),(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)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(4,5)],6)
=> ([(0,2),(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)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,5),(3,4)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(1,2),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(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),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,5),(1,4),(2,3)],6)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> ([(2,5),(3,4)],6)
=> 2 = 4 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(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)
=> ([],6)
=> 1 = 3 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(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)],6)
=> 2 = 4 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2 = 4 - 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 2
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ([],1)
=> 1 = 3 - 2
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(2,3),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1 = 3 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1 = 3 - 2
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2 = 4 - 2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1 = 3 - 2
([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? = 3 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 4 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10)],12)
=> ?
=> ? = 3 - 2
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8)],10)
=> ?
=> ? = 4 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 2
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,4),(3,6),(4,5),(6,7)],8)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([],9)
=> ? = 3 - 2
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 4 - 2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? = 3 - 2
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)
=> ? = 4 - 2
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,11),(2,3),(2,5),(2,8),(2,10),(3,4),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,10),(6,7),(6,11),(7,8),(9,10),(9,11),(10,11)],12)
=> ?
=> ? = 4 - 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ?
=> ? = 4 - 2
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St001349
Values
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> 0 = 3 - 3
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> 0 = 3 - 3
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 4 - 3
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],1)
=> ([],1)
=> 0 = 3 - 3
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(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)
=> 1 = 4 - 3
([(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)
=> 0 = 3 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 1 = 4 - 3
([],6)
=> ([(0,1),(0,2),(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)
=> ([],1)
=> ([],1)
=> 0 = 3 - 3
([(4,5)],6)
=> ([(0,2),(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)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(2,5),(3,4)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(2,5),(3,4),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(1,2),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 4 - 3
([(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),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(0,5),(1,4),(2,3)],6)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(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)
=> ([(2,5),(3,4)],6)
=> 1 = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(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)
=> ([],6)
=> 0 = 3 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(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)],6)
=> 1 = 4 - 3
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 4 - 3
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 1 = 4 - 3
([(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)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 1 = 4 - 3
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 3
([],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([],1)
=> ([],1)
=> 0 = 3 - 3
([(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(3,6),(4,5)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(2,3),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(2,6),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(1,2),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 4 - 3
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 3 - 3
([(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0 = 3 - 3
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ([(0,1),(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 1 = 4 - 3
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0 = 3 - 3
([(1,6),(2,5),(3,4)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? = 3 - 3
([(0,3),(1,2),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 3
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 4 - 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10)],12)
=> ?
=> ? = 3 - 3
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 3 - 3
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,5),(0,6),(0,7),(0,9),(1,2),(1,3),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 4 - 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 4 - 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8)],10)
=> ?
=> ? = 4 - 3
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,6),(1,2),(1,3),(1,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 3
([(0,1),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(2,4),(3,6),(4,5),(6,7)],8)
=> ? = 4 - 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,4),(0,5),(0,7),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 3
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([],9)
=> ? = 3 - 3
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(2,7),(3,7),(4,6),(5,6)],8)
=> ? = 4 - 3
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 4 - 3
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([],8)
=> ? = 3 - 3
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8)
=> ? = 4 - 3
([(0,1),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 4 - 3
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 3
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4 - 3
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? = 4 - 3
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,6),(1,8),(1,11),(2,3),(2,5),(2,8),(2,10),(3,4),(3,8),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,7),(5,10),(6,7),(6,11),(7,8),(9,10),(9,11),(10,11)],12)
=> ?
=> ? = 4 - 3
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ?
=> ? = 4 - 3
Description
The number of different graphs obtained from the given graph by removing an edge.
The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000313The number of degree 2 vertices of a graph. St001765The number of connected components of the friends and strangers graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000261The edge connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001645The pebbling number of a connected graph. St001621The number of atoms of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001316The domatic number of a graph.
Sorry, this statistic was not found in the database
or
add this statistic to the database ā it's very simple and we need your support!