|
%I #6 Dec 10 2013 12:23:46
%S 1,4,19,95,475,2376,11881,59406,297029,1485144,7425719,37128595,
%T 185642975,928214876,4641074381,23205371904,116026859520,580134297600,
%U 2900671488000,14503357440000,72516787200000,362583936000000
%N Dimension of 5-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 5 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).
%D C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
%D 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.
%H N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="http://arxiv.org/abs/math.CO/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math.CO/0502082, Canad. J. Math. 60 (2008), no. 2, 266-296.
%F G.f.: (1-q)*(1-q^2)*(1-q^3)*(1-q^4)*(1-q^5)/(1-5*q) a(n) = 23205371904*5^(n-15) for n>14
%e a(1) = 4 because x1 - x2, x2 - x3, x3 - x4, x4 - x5 are all killed by d_x1+d_x2+d_x3+d_x4+d_x5
%p coeffs(convert(series(mul(1-q^i,i=1..5)/(1-5*q),q,20),`+`)-O(q^20),q);
%Y Cf. A118266, A122369, A122391, A122392, A122393.
%K nonn,easy
%O 0,2
%A _Mike Zabrocki_, Aug 31 2006
|
