A122370 Dimension of 6-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 6 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).

1, 5, 29, 172, 1026, 6134, 36712, 219847, 1316963, 7890594, 47282065, 283344410, 1698058817, 10176618298, 60990528210, 365532989831, 2190756912988, 13129979193808, 78692862940748, 471636719623539

Offset: 0

Author

Mike Zabrocki, Aug 30 2006

Keywords

• nonn

Examples

			a(1) = 5 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6.
		

References

• C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
• M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.

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
• Index entries for linear recurrences with constant coefficients, signature (15,-81,192,-189,53)

Crossrefs

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122371, A122372.

Programs

Maple

coeffs(convert(series((1-10*q+35*q^2-50*q^3+24*q^4)/ (1-15*q+81*q^2 -192*q^3+189*q^4 -53*q^5),q,20), `+`) -O(q^20),q)

Mathematica

LinearRecurrence[{15, -81, 192, -189, 53}, {1, 5, 29, 172, 1026}, 20] (* Jean-François Alcover, Sep 22 2017 *)

Formula

o.g.f. (1-10*q+35*q^2-50*q^3+24*q^4) / (1-15*q+81*q^2 -192*q^3+189*q^4 -53*q^5) more generally, sum( n!/(n-d)!*q^d/prod((1-r*q),r=1..d), d=0..n) / sum( q^d/prod((1-r*q),r=1..d), d=0..n) where n=6.