A357257 -id:A357257 - cp's OEIS frontend

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

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

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.

Cf. A357242, A357248, A357257.

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).

A357248 Number of n-node tournaments that have exactly four circular triads.

Original entry on oeis.org

280, 6240, 75600, 954240, 12579840, 175392000, 2594592000, 40721049600, 677053977600, 11901451161600, 220690229760000, 4307253350400000, 88289523818496000, 1896762491559936000, 42625344258072576000, 1000193047805952000000, 24463730767033958400000, 622724156293184225280000

Offset: 5

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

nonn

			For n=5, the a(5)=280 solution is 5!*((7/3)*(5-4)+4*(5-5)+(7/6)(5-6)(5-7)[5>5]+(1/18)*(5-7)(5-8)(5-9)[5>6]+(1/1944)[5>7]*(5-8)!/(5-12)!)=5!*(7/3)*(5-4)=280.

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.

Cf. A357242, A357257, A357266.

CoefficientList[Series[(x^7-27*x^6+216*x^5-702*x^4+972*x^3-405*x^2-243*x+189)*x^5/((3^4)*(1-x)^5), {x,0,22}], x]Table[n!, {n,0,22}] (* Stefano Spezia, Sep 27 2022 *)

a(n) = n!*((7/3)*(n-4)+4*(n-5)+(7/6)(n-6)(n-7)[n>5]+(1/18)*(n-7)(n-8)(n-9)[n>6]+(1/1944)[n>7]*(n-8)!/(n-12)!) (see Kadane).

E.g.f.: (x^7-27*x^6+216*x^5-702*x^4+972*x^3-405*x^2-243*x+189)*x^5/((3^4)*(1-x)^5).

cp's OEIS Frontend

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

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