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A357248 Number of n-node tournaments that have exactly four circular triads. 2
280, 6240, 75600, 954240, 12579840, 175392000, 2594592000, 40721049600, 677053977600, 11901451161600, 220690229760000, 4307253350400000, 88289523818496000, 1896762491559936000, 42625344258072576000, 1000193047805952000000, 24463730767033958400000, 622724156293184225280000 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,1

LINKS

Table of n, a(n) for n=5..22.

J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.

FORMULA

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).

EXAMPLE

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.

MATHEMATICA

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 *)

CROSSREFS

Cf. A357242, A357257, A357266.

Sequence in context: A187384 A297724 A024214 * A094895 A223107 A218411

Adjacent sequences: A357245 A357246 A357247 * A357249 A357250 A357251

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified November 30 19:14 EST 2022. Contains 358453 sequences. (Running on oeis4.)