- FindStat

Identifier

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
St000749: Integer partitions ⟶ ℤ

Values

([],2) => ([],2) => ([(0,1)],2) => [1] => 0

([],3) => ([],3) => ([(0,1),(0,2),(1,2)],3) => [3] => 0

([(1,2)],3) => ([(1,2)],3) => ([(0,2),(1,2)],3) => [1,1] => 0

([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => ([(1,2)],3) => [1] => 0

([],4) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => 0

([(2,3)],4) => ([(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => 0

([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => [3,1] => 0

([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => [3] => 0

([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => [4] => 0

([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => [1,1,1] => 0

([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => [1,1,1] => 0

([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => [1,1] => 0

([],5) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [10] => 0

([(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) => [9] => 0

([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [8] => 0

([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6,1] => 1

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => 0

([(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) => [8] => 0

([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [7] => 0

([(0,1),(2,4),(3,4)],5) => ([(0,1),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [7] => 0

([(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) => [7] => 0

([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5,1] => 0

([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5,1] => 0

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => 0

([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => 0

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => 0

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => 0

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => 1

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => [3,1] => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => [3] => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(3,4)],5) => [3] => 0

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [6] => 0

([(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) => [6] => 0

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [4,1] => 1

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => [4] => 0

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => 0

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => [1,1,1] => 0

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => [1,1,1,1] => 0

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => [1,1,1] => 0

([(2,5),(3,5),(4,5)],6) => ([(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) => [12] => 0

([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10,1] => 1

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 0

([(2,5),(3,4),(4,5)],6) => ([(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) => [12] => 0

([(1,2),(3,5),(4,5)],6) => ([(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) => [12] => 0

([(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) => [12] => 0

([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [11] => 0

([(0,1),(2,5),(3,5),(4,5)],6) => ([(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) => [11] => 0

([(2,5),(3,4),(3,5),(4,5)],6) => ([(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) => [11] => 0

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9,1] => 0

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9,1] => 0

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => 0

([(2,4),(2,5),(3,4),(3,5)],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) => [9] => 0

([(0,5),(1,5),(2,4),(3,4)],6) => ([(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) => [11] => 0

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8,1] => 1

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [10] => 0

([(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) => [9] => 0

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 0

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 0

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8] => 0

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8,1] => 1

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8,1] => 1

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8] => 0

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1,1] => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1,1] => 0

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => 1

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1,1] => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1,1] => 0

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6,1] => 1

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 0

([(0,4),(0,5),(1,4),(1,5),(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) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 0

([(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) => [12] => 0

([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [11] => 0

([(0,1),(2,5),(3,4),(4,5)],6) => ([(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) => [11] => 0

([(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) => [11] => 0

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [10] => 0

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 0

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(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) => [10] => 0

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8,1] => 1

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => [8,1] => 1

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => [8] => 0

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [10] => 0

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => 0

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => 0

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [9] => 0

([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7,1] => 0

([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [9] => 0

([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 0

([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [7,1] => 0

([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [1,1,1,1,1] => 0

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 0

([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [10] => 0

([(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) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [8] => 0

([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(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) => [10] => 0

([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [9] => 0

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [9] => 0

([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => [9] => 0

([(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) => [8] => 0

([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [9] => 0

([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [7] => 0

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => [7,1] => 0

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => [7,1] => 0

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search for individual values

searching the database for the individual values of this statistic

Description

The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].

Map

to edge-partition of biconnected components

Description

Sends a graph to the partition recording the number of edges in its biconnected components.
The biconnected components are also known as blocks of a graph.

Map

Ore closure

Description

The Ore closure of a graph.
The Ore closure of a connected graph $G$ has the same vertices as $G$, and the smallest set of edges containing the edges of $G$ such that for any two vertices $u$ and $v$ whose sum of degrees is at least the number of vertices, then $(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.

Map

complement

Description

The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.