A013976 - OEIS

%I #20 Sep 09 2022 22:27:19

%S 1,2,6,32,600,20544,1218224,160241152,42129744768,21293228876800,

%T 22220602090444032,45959959305969143808,188702851949391611599872,

%U 1566370829898884830446395392,25801614282517307546368494213120,851300764257208779000509484701188096

%N Number of tournaments on n nodes with a unique winner.

%H Andrew Howroyd, <a href="/A013976/b013976.txt">Table of n, a(n) for n = 1..20</a>

%H Yaakov Malinovsky and John W. Moon, <a href="https://arxiv.org/abs/2208.14932">On Round-Robin Tournaments with a Unique Maximum Score and Some Related Results</a>, arXiv:2208.14932 [math.CO], 2022.

%H <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>

%o (PARI)

%o Winners(n)={

%o   local(M=Map(Mat([x^0, 1])));

%o   my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));

%o   my(merge(r, p, v)=acc(p + sum(i=1, poldegree(p)-r-1, polcoef(p,i)*(1-x^i)), v));

%o   my(recurse(r, p, i, q, v, e)=if(i<0, merge(r, x^e+q, v), my(t=polcoef(p, i)); for(k=0, t, self()(r, p, i-1, (k+x*(t-k))*x^i+q, binomial(t, k)*v, e+k))));

%o   for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(n-k, p, poldegree(p), 0, src[i, 2], 0)));

%o   Mat(M);

%o }

%o a(n)={my(M=Winners(n)); sum(i=1, matsize(M)[1], if(pollead(M[i,1])==1, M[i,2]))} \\ _Andrew Howroyd_, Feb 29 2020

%Y Cf. A125031.

%K nonn,nice

%O 1,2

%A Michael Stob (stob(AT)udu.calvin.edu)

%E a(1)=1 inserted and a(12) and beyond from _Andrew Howroyd_, Feb 28 2020