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

0,2

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

Table of n, a(n) for n=0..23.

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.

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

Example

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

Maple

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

Crossrefs

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

Sequence in context: A345670 A370998 A239392 * A012880 A200613 A149073

Adjacent sequences: A122390 A122391 A122392 * A122394 A122395 A122396

Keyword

nonn,easy

Author

Mike Zabrocki, Aug 31 2006

Status

approved