A122372 Dimension of 8-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 8 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, 7, 55, 438, 3498, 27962, 223604, 1788406, 14305102, 114429193, 915366442, 7322521512, 58577537621, 468602617723, 3748697751384, 29988696932490, 239903055854075, 1919175464438065, 15353030007717639, 122821355074655309

Offset

0,2

Links

Table of n, a(n) for n=0..19.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.

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.

Index entries for linear recurrences with constant coefficients, signature (28,-316,1845,-5925,10190,-8249,2119).

Formula

G.f.: (1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/ (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7) 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=8.

Example

A122371 a(1) = 7 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7, x7-x8 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.

Maple

coeffs(convert(series((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/ (1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7), q, 20), `+`)-O(q^20), q);

Mathematica

n = 8; gf = 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}] + O[q]^20;
CoefficientList[gf, q] (* Jean-François Alcover, Dec 03 2018 *)

Crossrefs

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122371.

Sequence in context: A015564 A070997 A370177 * A342935 A083068 A362299

Adjacent sequences: A122369 A122370 A122371 * A122373 A122374 A122375

Keyword

nonn

Author

Mike Zabrocki, Aug 30 2006

Status

approved