A122371 Dimension of 7-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 7 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, 6, 41, 285, 1989, 13901, 97215, 680079, 4758408, 33297267, 233014444, 1630701426, 11412409945, 79870754268, 558989013403, 3912210491549, 27380636068267, 191631324294463, 1341190961828143, 9386756237545989

Offset: 0

Author

Mike Zabrocki, Aug 30 2006

Keywords

• nonn

Examples

			a(1) = 6 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6+d_x7.
		

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 (21,-170,669,-1314,1157,-309).

Crossrefs

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122372.

Programs

Maple

coeffs(convert(series((1-15*q+ 85*q^2-225*q^3+274*q^4-120*q^5) / (1-21*q+170*q^2-669*q^3+1314*q^4-1157*q^5+309*q^6),q,20),`+`)-O(q^20),q);

Mathematica

LinearRecurrence[{21, -170, 669, -1314, 1157, -309}, {1, 6, 41, 285, 1989, 13901}, 20] (* Jean-François Alcover, Sep 22 2017 *)

Formula

G.f.: (1-15*q+ 85*q^2-225*q^3+274*q^4-120*q^5) / (1-21*q+170*q^2-669*q^3 +1314*q^4-1157*q^5 +309*q^6) 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=7.