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%I #8 Dec 04 2012 12:01:34
%S 1,20,243,3104,46625,823500,16777159,387420416,9999999909,
%T 285311670500,8916100448123,302875106592096,11112006825557833,
%U 437893890380859164,18446744073709551375,827240261886336763904,39346408075296537575117
%N a(n) = degree in N of the number of orbits under S_N of the set of n-tuples of partitions of {1,...,N} into n subsets.
%C This formula and its first two values are given in Bergman, p. 18. Abstract: Erdos and Szekeres showed in 1978 that for any four positive integers satisfying m_1+m_2 = n_1+n_2, the two binomial coefficients (m_1+m_2)!/m_1! m_2! and (n_1+n_2)!/n_1! n_2! have a common divisor >1. The analogous statement for families of k k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman. Erdos and Szekeres remark that if m_1, m_2, n_1, n_2 as above are all >1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m_1+m_2. Such a bound is here obtained.
%C Results are proved that narrow the class of possible counterexamples to Wasserman's conjecture. On the other hand, several plausible generalizations of that conjecture are shown to be false.
%H George M. Bergman, <a href="http://arXiv.org/abs/0806.0607">On common divisors of multinomial coefficients</a>, arXiv:0806.0607 Jun 03, 2008.
%H George M. Bergman, <a href="http://math.berkeley.edu/~gbergman/papers/unpub/from_nomial.pdf">Addenda to "On common divisors of multinomial coefficients"</a>
%F a(n) = (n^n) - 1 - n*(n-1). a(n) = A000312(n) - 1 - n*(n-1).
%t Table[n^n-1-n(n-1),{n,2,20}] (* _Harvey P. Dale_, Dec 04 2012 *)
%Y Cf. A000312.
%K easy,nonn
%O 2,2
%A _Jonathan Vos Post_, Jun 04 2008
%E Definition clarified by George M. Bergman, Jul 05 2010