A122369 Dimension of 5-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 5 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, 4, 19, 93, 459, 2273, 11274, 55964, 277924, 1380527, 6858356, 34074280, 169297743, 841173845, 4179517118, 20766807551, 103184684826, 512698227699, 2547469553647, 12657750705603, 62893284231103, 312501512711984, 1552744642741738, 7715214279423070

Offset: 0

Author

Mike Zabrocki, Aug 30 2006

Keywords

• nonn

Examples

			a(1) = 4 because x1-x2, x2-x3, x3-x4, x4-x5 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5.
		

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

Crossrefs

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.

Programs

Maple

coeffs(convert(series((1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4),q,30),`+`)-O(q^30),q);

Mathematica

gf = With[{n = 5}, Sum[n!/(n-d)! q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}]/Sum[ q^d/Product[(1 - r q), {r, 1, d}], {d, 0, n}]]; CoefficientList[gf + O[q]^22, q] (* Jean-François Alcover, Nov 17 2018 *)

Formula

G.f. (1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4) 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=5.