proposed

approved

editing

proposed

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.

approved

editing

proposed

approved

editing

proposed

Michael Stob (stob(AT)udu.Calvincalvin.EDUedu)

proposed

editing

editing

proposed

1, 2, 6, 32, 600, 20544, 1218224, 160241152, 42129744768, 21293228876800, 22220602090444032, 45959959305969143808, 188702851949391611599872, 1566370829898884830446395392, 25801614282517307546368494213120, 851300764257208779000509484701188096

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

TournamentsByScoreSeqWinners(n, isok)={

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

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

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

my(recurse(r, p, i, q, v, e)=if(i<0, if(isokmerge(x^e+q, r), acc(, 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))));

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)));

a(n)={vecsummy(TournamentsByScoreSeqM=Winners(n, )); sum(i=1, matsize(p, rM)->r>0||[1], if(pollead(pM[i, 1])==1), M[, i, 2]))} \\ Andrew Howroyd, Feb 28 29 2020

a(1)=1 inserted and a(12)-a(15) and beyond from Andrew Howroyd, Feb 28 2020

TournamentsByScoreSeq(n, fokisok)={

my(recurse(r, p, i, q, v, e)=if(i<0, if(fokisok(x^e+q, r), acc(x^e+q, v)), my(t=polcoeffpolcoef(p, i)); for(k=0, t, self()(r, p, i-1, (k+x*(t-k))*x^i+q, binomial(t, k)*v, e+k))));

1, 2, 6, 32, 600, 20544, 1218224, 160241152, 42129744768, 21293228876800, 22220602090444032, 45959959305969143808, 188702851949391611599872, 1566370829898884830446395392, 25801614282517307546368494213120

a(1)=1 inserted and a(12)-a(1415) from Andrew Howroyd, Feb 28 2020