A357266 Number of n-node tournaments that have exactly five circular triads.

24, 3648, 90384, 1304576, 19958400, 311592960, 5054353920, 85709352960, 1523221539840, 28387834675200, 554575551129600, 11345938174771200, 242796629621145600, 5427273747293798400, 126546947417899008000

Offset

5,1

Links

Table of n, a(n) for n=5..19.

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

Kadane proves that a(n) = n!*((1/5)*(n-4)+(14/3)*(n-5)+8*(n-6)I(n>5)+(7/9)*(n-6)*(n-7)I(n>5)+(10/3)*(n-7)*(n-8)I(n>6)+(5/18)*(n-8)*(n-9)*(n-10)I(n>7)+(1/162)*(n-9)*(n-10)*(n-11)*(n-12)I(n>8)+(1/29160)*(n-10)*(n-11)*(n-12)*(n-13)*(n-14)I(n>9)), where I(p) is the indicator function: 1 if p is true and 0 otherwise.

E.g.f.: (5*x^10-180*x^9+2205*x^8-12150*x^7+34155*x^6-51840*x^5+38313*x^4-3942*x^3-11502*x^2+4698*x+243)*x^5/(5*3^5*(1-x)^6).

Crossrefs

Cf. A357242, A357248, A357257.

Sequence in context: A275637 A223123 A278651 * A166768 A326120 A159399

Adjacent sequences: A357263 A357264 A357265 * A357267 A357268 A357269

Keyword

nonn,easy

Author

Ian R Harris, Ryan P. A. McShane, Sep 22 2022

Status

approved