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 A000571 Number of different score sequences that are possible in an n-team round-robin tournament. (Formerly M1189 N0459) 19
 1, 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, 158808, 531469, 1799659, 6157068, 21258104, 73996100, 259451116, 915695102, 3251073303, 11605141649, 41631194766, 150021775417, 542875459724, 1972050156181, 7189259574618, 26295934251565, 96478910768821, 354998461378719, 1309755903513481 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A tournament is a complete graph with one arrow on each edge; the score of a node is its out-degree; a(n) is number of different score sequences when there are n nodes. Also number of nonnegative integer points (p_1, p_2, ..., p_n) in polytope p_0=p_{n+1}=0, 2p_i -(p_{i+1}+p_{i-1}) <= 1, p_i >= 0, i=1, ..., n. Also number of score sequences of length n: weakly increasing sequences a[0,1,...,n-1] where sum(j=0..k, a[j]) >= k*(k+1)/2 and sum(j=0..n-1, a[j]) = (n+1)*n/2; see example. - Joerg Arndt, Mar 29 2014 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 21. P. A. MacMahon, An American tournament treated by the calculus of symmetric functions, Quart. J. Pure Appl. Math., 49 (1920), 1-36. [Gives a[0)-a(8). - N. J. A. Sloane, Jun 11 2016) J. W. Moon, Topics on Tournaments. Holt, NY, 1968, p. 68 (but table contains errors). N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1675 (terms 0..100 from Sean A. Irvine) C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005. David E. Brown, Eric Culver, Bryce Frederickson, Sidney Tate, Brent J. Thomas, Entropy of Tournament Digraphs, arXiv:1812.09458 [math.CO], 2018. Sebrina Ruth Cropper, Ranking Score Vectors of Tournaments (2011). All Graduate Reports and Creative Projects. Paper 91. Utah State University, School of Graduate Studies. P. A. MacMahon, An American tournament treated by the calculus of symmetric functions, Quart. J. Pure Appl. Math., 49 (1920), 1-36. Gives a(1) to a(9). [Annotated scanned copy, scanned at 300 dpi. Do not replace with a smaller file as the print is very tiny and hard to read.] John W. Moon, Topics on tournaments, Holt, Rinehard and Winston (1968), see page 88. T. V. Narayana and D. H. Best, Computation of the number of score sequences in round-robin tournaments, Canad. Math. Bull., 7 (1964), 133-136 (but table contains errors). D. Recoskie and J. Sawada, The Taming of Two Alley CATs, 2012. J. Riordan, The number of score sequences in tournaments, J. Combin. Theory, 5 (1968), 87-89. [The main result of this paper seems to be wrong - see A210726.] Eric Weisstein's World of Mathematics, Score Sequence. K. Winston, Letter to N. J. A. Sloane, Aug 05 1978 FORMULA Let f_1(T, E)=1 if T=E>=0, =0 else; f_n(T, E)=0 if T-E= 2. Riordan seems to claim that a(n) = 2*a(n-1)-a(n-2) + A046919(n), but this is not true. - N. J. A. Sloane, May 06 2012 Logarithmic derivative yields A145855, the number of n-member subsets of 1..2n-1 whose elements sum to a multiple of n. - Paul D. Hanna, Jul 17 2013 a(n) = (1/n) * Sum_{k=1..n} A145855(k) * a(n-k) for n>0 with a(0)=1. - Paul D. Hanna, Jul 17 2013 EXAMPLE a(3)=2, since either one node dominates [ 2,1,0 ] or each node defeats the next [ 1,1,1 ]. G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 22*x^6 + 59*x^7 + 167*x^8 +... where the logarithm begins: log(A(x)) = x + x^2/2 + 4*x^3/3 + 9*x^4/4 + 26*x^5/5 + 76*x^6/6 +...+ A145855(n)*x^n/n +... From Joerg Arndt, Mar 29 2014: (Start) The a(6) = 22 score sequences of length 6 are: 01:  [ 0 1 2 3 4 5 ] 02:  [ 0 1 2 4 4 4 ] 03:  [ 0 1 3 3 3 5 ] 04:  [ 0 1 3 3 4 4 ] 05:  [ 0 2 2 2 4 5 ] 06:  [ 0 2 2 3 3 5 ] 07:  [ 0 2 2 3 4 4 ] 08:  [ 0 2 3 3 3 4 ] 09:  [ 0 3 3 3 3 3 ] 10:  [ 1 1 1 3 4 5 ] 11:  [ 1 1 1 4 4 4 ] 12:  [ 1 1 2 2 4 5 ] 13:  [ 1 1 2 3 3 5 ] 14:  [ 1 1 2 3 4 4 ] 15:  [ 1 1 3 3 3 4 ] 16:  [ 1 2 2 2 3 5 ] 17:  [ 1 2 2 2 4 4 ] 18:  [ 1 2 2 3 3 4 ] 19:  [ 1 2 3 3 3 3 ] 20:  [ 2 2 2 2 2 5 ] 21:  [ 2 2 2 2 3 4 ] 22:  [ 2 2 2 3 3 3 ] (End) MATHEMATICA max = 40; (* b = A145855 *) b = 1; b[n_] := DivisorSum[n, (-1)^(n+#)* EulerPhi[n/#]*Binomial[2*#, #]/(2*n)&]; s = Exp[Sum[b[m]*x^m/m, {m, 1, max}]] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 06 2015, adapted from PARI *) PROG (PARI) {A145855(n)=sumdiv(n, d, (-1)^(n+d)*eulerphi(n/d)*binomial(2*d, d)/(2*n))} {a(n)=polcoeff(exp(sum(m=1, n, A145855(m)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 17 2013 CROSSREFS Cf. A007747, A210726, A145855. See A274098 for the most likely score sequence. Sequence in context: A177377 A321994 A035053 * A077003 A210726 A046917 Adjacent sequences:  A000568 A000569 A000570 * A000572 A000573 A000574 KEYWORD nonn,nice AUTHOR EXTENSIONS a(11) corrected by Kenneth Winston, Aug 05 1978 More terms from David W. Wilson STATUS approved

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Last modified October 19 19:44 EDT 2021. Contains 348091 sequences. (Running on oeis4.)