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.

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.

Links

Table of n, a(n) for n=2..18.

George M. Bergman, On common divisors of multinomial coefficients, arXiv:0806.0607 Jun 03, 2008.

George M. Bergman, Addenda to "On common divisors of multinomial coefficients"

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

Adjacent sequences: A140121 A140122 A140123 * A140125 A140126 A140127

Keyword

easy,nonn

Author

Jonathan Vos Post, Jun 04 2008

Extensions

Definition clarified by George M. Bergman, Jul 05 2010

Status

approved