A350608 Number of weakly connected subgraphs of the transitive tournament on {1,...,n}.

1, 1, 4, 31, 474, 14357, 865024, 103931595, 24935913222, 11956100981537, 11460773522931212, 21967828926423843319, 84207961512578582993810, 645554571594493917538073933, 9897742810470352880099047702936, 303505765229448690912596327628571427

Offset: 1

Author

Don Knuth, Jan 16 2022

Keywords

• nonn

Comments

The transitive tournament on n labeled nodes 1, ..., n has n*(n-1)/2 arcs, namely i->j for 1 <= i < j <= n.

Examples

			a(4)=31: the 31 weakly connected subgraphs when n=4 are the 1+6+15 digraphs that have only 0 or 1 or 2 arcs, plus the four digraphs with three arcs that leave one vertex untouched, plus the five digraphs with three arcs that make an N:
  1->3,1->4,2->3;
  1->3,1->4,2->4;
  1->3,2->3,2->4;
  1->4,2->3,2->4;
  1->2,1->4,3->4.
		

References

• Jean Francois Pacault, "Computing the weak components of a
• directed graph," SIAM Journal on Computing 3 (1974), 56-61.

Links

• R. L. Graham, D. E. Knuth, and T. S. Motzkin, Complements and transitive closures, Discrete Mathematics 2 (1972), 17--29.
• Don Knuth, Weak Components Revived, January 2022.
• Don Knuth, Pre-Fascicle 12A of TAOCP, Volume 4, January 2022.

Crossrefs

Cf. A350609, A350610.