Ryan P. A. McShane - cp's OEIS frontend

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

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

nonn

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

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, A357266.

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 *)

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).

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).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

User: Ryan P. A. McShane

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Formula