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).

1, 3, 11, 44, 176, 706, 2824, 11296, 45183, 180731, 722925, 2891700, 11566800, 46267200, 185068800, 740275200, 2961100800, 11844403200, 47377612800, 189510451200, 758041804800, 3032167219200, 12128668876800, 48514675507200

Offset: 0

Author

Mike Zabrocki, Aug 31 2006

Keywords

• nonn
• easy

Examples

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

References

• C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
• 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.

Links

• N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082, Canad. J. Math. 60 (2008), no. 2, 266-296.

Crossrefs

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

Programs

Maple

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

Formula

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