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A140124
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.
0
1, 20, 243, 3104, 46625, 823500, 16777159, 387420416, 9999999909, 285311670500, 8916100448123, 302875106592096, 11112006825557833, 437893890380859164, 18446744073709551375, 827240261886336763904, 39346408075296537575117
OFFSET
2,2
COMMENTS
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.
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.
FORMULA
a(n) = (n^n) - 1 - n*(n-1). a(n) = A000312(n) - 1 - n*(n-1).
MATHEMATICA
Table[n^n-1-n(n-1), {n, 2, 20}] (* Harvey P. Dale, Dec 04 2012 *)
CROSSREFS
Cf. A000312.
Sequence in context: A040075 A138442 A341196 * A123954 A125432 A278673
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Jun 04 2008
EXTENSIONS
Definition clarified by George M. Bergman, Jul 05 2010
STATUS
approved