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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).
3
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
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
KEYWORD
nonn,easy
AUTHOR
Mike Zabrocki, Aug 31 2006
STATUS
approved