Identifier
Values
{{1}} => [1] => ([],1) => ([(0,1)],2) => 1
{{1,2}} => [2,1] => ([(0,1)],2) => ([(0,1),(0,2),(1,2)],3) => 2
{{1},{2}} => [1,2] => ([],2) => ([(0,2),(1,2)],3) => 1
{{1,2,3}} => [2,3,1] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
{{1,2},{3}} => [2,1,3] => ([(1,2)],3) => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
{{1,3},{2}} => [3,2,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
{{1},{2,3}} => [1,3,2] => ([(1,2)],3) => ([(0,3),(1,2),(1,3),(2,3)],4) => 2
{{1},{2},{3}} => [1,2,3] => ([],3) => ([(0,3),(1,3),(2,3)],4) => 1
{{1,2,3,4}} => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1,2,3},{4}} => [2,3,1,4] => ([(1,3),(2,3)],4) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1,2,4},{3}} => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,2},{3,4}} => [2,1,4,3] => ([(0,3),(1,2)],4) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 3
{{1,2},{3},{4}} => [2,1,3,4] => ([(2,3)],4) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1,3,4},{2}} => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,3},{2,4}} => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2
{{1,3},{2},{4}} => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1,4},{2,3}} => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
{{1},{2,3,4}} => [1,3,4,2] => ([(1,3),(2,3)],4) => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1},{2,3},{4}} => [1,3,2,4] => ([(2,3)],4) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1,4},{2},{3}} => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2,4},{3}} => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
{{1},{2},{3,4}} => [1,2,4,3] => ([(2,3)],4) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2
{{1},{2},{3},{4}} => [1,2,3,4] => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
{{1,2,3,4,5}} => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,2,3,5},{4}} => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,3},{4,5}} => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,3},{4},{5}} => [2,3,1,4,5] => ([(2,4),(3,4)],5) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,4},{3},{5}} => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2},{3,4,5}} => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1,2},{3,5},{4}} => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,2},{3},{4,5}} => [2,1,3,5,4] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => ([(3,4)],5) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,3,4,5},{2}} => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,3,4},{2},{5}} => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,3},{2,4},{5}} => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1,4,5},{2,3}} => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1,4},{2,3},{5}} => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1},{2,3,4,5}} => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1},{2,3,4},{5}} => [1,3,4,2,5] => ([(2,4),(3,4)],5) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2,3},{4,5}} => [1,3,2,5,4] => ([(1,4),(2,3)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => ([(3,4)],5) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2,4,5},{3}} => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2,4},{3,5}} => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 2
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2,5},{3,4}} => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
{{1},{2},{3,4,5}} => [1,2,4,5,3] => ([(2,4),(3,4)],5) => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => ([(3,4)],5) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => ([(3,4)],5) => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1
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Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
cone
Description
The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.