A357257 Number of n-node tournaments that have exactly three circular triads.

240, 2880, 33600, 403200, 5093760, 68275200, 972787200, 14724864000, 236396160000, 4016659046400, 72067387392000, 1362306097152000, 27071765360640000, 564357385912320000, 12317692759916544000, 280955128203509760000

Offset: 5

Author

Ian R Harris, Ryan P. A. McShane, Sep 20 2022

Keywords

• nonn

Examples

			a(6) = 6!*(2*(6-4) + (1/3)*(6-5)*(6-6) + (1/162)*(6-6)*(6-7)*(6-8)*[6>5]) = 2880.
		

Links

• Ian R. Harris and Ryan P. A. McShane, Counting Tournaments with a Specified Number of Circular Triads, Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.7. See pages 2, 23.
• J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.

Crossrefs

Cf. A357242, A357248, A357266.

Programs

Mathematica

Table[n!*(2*(n-4) + (1/3)*(n-5)*(n-6) + (1/162)*(n-6)*(n-7)*(n-8)*Boole[n>5]), {n,5,20}] (* Stefano Spezia, Sep 27 2022 *)

Formula

a(n) = n!*(2*(n-4) + (1/3)*(n-5)*(n-6) + (1/162)*(n-6)*(n-7)*(n-8)*[n>5]) (see Kadane).

E.g.f.: (x^4 - 18*x^3 + 72*x^2 - 108*x + 54)*x^5/((3^3)*(1-x)^4).