%I #26 Jul 06 2024 10:30:31

%S 0,0,0,0,0,0,10,70,280,840,2100,4620,9240,17160,30030,50050,80080,

%T 123760,185640,271320,387600,542640,746130,1009470,1345960,1771000,

%U 2302300,2960100,3767400,4750200,5937750,7362810,9061920,11075680,13449040,16231600,19477920

%N Number of different 3 against 3 matches given n players.

%C Given n players there are a(n) different ways of arranging those players in a 3 against 3 contest.

%C Number of ways to select two disjoint subsets of size 3 from a set of n elements. - _Joerg Arndt_, Mar 29 2016

%H Colin Barker, <a href="/A271040/b271040.txt">Table of n, a(n) for n = 0..1000</a>

%H Sela Fried, <a href="https://arxiv.org/abs/2406.18923">Counting r X s rectangles in nondecreasing and Smirnov words</a>, arXiv:2406.18923 [math.CO], 2024. See p. 9.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/72.

%F a(n) = binomial(n,3) * binomial(n-3,3) / 2. - _Joerg Arndt_, Mar 29 2016

%F From _Colin Barker_, Mar 29 2016: (Start)

%F a(n) = 10*A000579(n).

%F a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>6.

%F G.f.: 10*x^6 / (1-x)^7.

%F (End)

%e When there are 6 players, there are 10 different 3 against 3 matches that can be played: ABC v DEF, ABD v CEF, ABE v CDF, ABF v CDE, ACD v BEF, ACE v BDF, ACF v BDE, ADE v BCF, ADF v BCE, AEF v BCD.

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,10},40] (* _Harvey P. Dale_, Sep 17 2016 *)

%o (PARI) concat(vector(6), Vec(10*x^6/(1-x)^7 + O(x^50))) \\ _Colin Barker_, Mar 29 2016

%o (PARI) a(n)=binomial(n,3)*binomial(n-3,3)/2 \\ _Charles R Greathouse IV_, May 22 2018

%Y Cf. A050534, the analogous situation for 2 against 2 matches.

%K nonn,easy

%O 0,7

%A _Elliott Line_, Mar 29 2016