0,3

For n>=4, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2, x_3, x_4 in {1,2,...,n} and fixed y_1, y_2, y_3, y_ 4 in {1,2,3,4} we have f(x_i)<>y_i, (i=1,2,3,4). - Milan R. Janjic (agnus(AT)blic.net), May 13 2007

N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, http://arXiv.org/abs/math.CO/0502082, to appear Canad. Math J.

C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

g.f. (1-q)^4/(1-4q) sum( (-1)^k*C(4,k) 4^(n-k); k=0..min(n,4));

the enveloping algebra of the derivied free Lie algebra is characterized as the interesection of the kernels of all partial derivative operators in the space of non-commutative polynomials, a(0) = 1 since all constants are killed by derivatives, a(1) = 0 since no polys of degree 1 are killed, a(2) = 6 since all Lie brackets [x1,x2], [x1,x3], [x1, x4], [x2,x3], [x2,x4], [x3,x4] are killed by all derivative operators

f:=n->add((-1)^k*C(4, k)*4^(n-k), k=0..min(n, 4)); seq(f(i), i=0..15);

Cf. A027377, A118264, A118266.

Sequence in context: A028402 A092760 A058494 this_sequence A036755 A045470 A117998

Adjacent sequences: A118262 A118263 A118264 this_sequence A118266 A118267 A118268

nonn

Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Apr 20 2006