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

5,1

Links

Table of n, a(n) for n=5..20.

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.

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).

Example

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

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 *)

Crossrefs

Cf. A357242, A357248, A357266.

Sequence in context: A004009 A269033 A180277 * A005950 A279746 A004536

Adjacent sequences: A357254 A357255 A357256 * A357258 A357259 A357260

Keyword

nonn

Author

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

Status

approved