A122393 - OEIS

A122393

Dimension of 4-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 4 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).

%I #6 Dec 10 2013 12:23:12

%S 1,3,11,44,176,706,2824,11296,45183,180731,722925,2891700,11566800,

%T 46267200,185068800,740275200,2961100800,11844403200,47377612800,

%U 189510451200,758041804800,3032167219200,12128668876800,48514675507200

%N Dimension of 4-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 4 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)/(1-4*q) a(n) = 722925*4^(n-10) for n>9

%e a(1) = 3 because x1 - x2, x2 - x3, x3 - x4 are all killed by d_x1+d_x2+d_x3+d_x4

%p coeffs(convert(series(mul(1-q^i,i=1..4)/(1-4*q),q,20),`+`)-O(q^20),q);

%Y Cf. A118265, A122368, A122391, A122392, A122394.

%K nonn,easy

%O 0,2

%A _Mike Zabrocki_, Aug 31 2006