A122392 Dimension of 3-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 3 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, 2, 5, 15, 46, 139, 416, 1248, 3744, 11232, 33696, 101088, 303264, 909792, 2729376, 8188128, 24564384, 73693152, 221079456, 663238368, 1989715104, 5969145312, 17907435936, 53722307808, 161166923424, 483500770272, 1450502310816

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..26.

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)/(1-3*q) 3^n - 3^(n-1) - 3^(n-2) + 3^(n-4) + 3^(n-5) - 3^(n-6) (for n>5) a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 15, a(4) = 46, a(5) = 139, a(n) = 416*3^(n-6) for n>5

Example

a(1) = 2 because x1 - x2, x2 - x3 are killed by d_x1 + d_x2 + d_x3
a(2) = 5 because x1 x2 - x2 x1, x1 x3 - x3 x1, x2 x3 - x3 x2, 2 x1 x2 - x2 x2 - 2 x1 x3 + x3 x3,
x1 x1 - 2 x2 x1 + 2 x2 x3 - x3 x3 are killed by d_x1 + d_x2 + d_x3, d_x1^2 + d_x2^2 + d_x3^2 and
d_x1 d_x2 + d_x1 d_x3 + d_x2 d_x3

Maple

coeffs(convert(series(mul(1-q^i, i=1..3)/(1-3*q), q, 20), `+`)-O(q^20), q);

Crossrefs

Cf. A118264, A122367, A122391, A122393, A122394.

Sequence in context: A065848 A148359 A287582 * A139782 A378339 A047086

Adjacent sequences: A122389 A122390 A122391 * A122393 A122394 A122395

Keyword

nonn,easy

Author

Mike Zabrocki, Aug 31 2006

Status

approved