cp's OEIS Frontend

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).

%I A122372 #19 Dec 03 2018 05:06:13
%S A122372 1,7,55,438,3498,27962,223604,1788406,14305102,114429193,915366442,
%T A122372 7322521512,58577537621,468602617723,3748697751384,29988696932490,
%U A122372 239903055854075,1919175464438065,15353030007717639,122821355074655309
%N 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).
%H A122372 N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="https://arxiv.org/abs/math/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
%H A122372 C. Chevalley, <a href="http://www.jstor.org/stable/2372597">Invariants of finite groups generated by reflections</a>, Amer. J. Math. 77 (1955), 778-782.
%H A122372 M. C. Wolf, <a href="http://dx.doi.org/10.1215/S0012-7094-36-00253-3">Symmetric functions of noncommutative elements</a>, Duke Math. J. 2 (1936), 626-637.
%H A122372 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-316,1845,-5925,10190,-8249,2119).
%F A122372 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.
%e A122372 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.
%p A122372 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);
%t A122372 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;
%t A122372 CoefficientList[gf, q] (* _Jean-François Alcover_, Dec 03 2018 *)
%Y A122372 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122371.
%K A122372 nonn
%O A122372 0,2
%A A122372 _Mike Zabrocki_, Aug 30 2006