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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). 5

%I

%S 1,6,41,285,1989,13901,97215,680079,4758408,33297267,233014444,

%T 1630701426,11412409945,79870754268,558989013403,3912210491549,

%U 27380636068267,191631324294463,1341190961828143,9386756237545989

%N 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).

%D C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.

%D M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.

%H N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, <a href="http://arxiv.org/abs/math.CO/0502082">Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables</a>, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-170,669,-1314,1157,-309).

%F 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.

%e 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.

%p 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);

%t LinearRecurrence[{21, -170, 669, -1314, 1157, -309}, {1, 6, 41, 285, 1989, 13901}, 20] (* _Jean-Fran├žois Alcover_, Sep 22 2017 *)

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

%K nonn

%O 0,2

%A _Mike Zabrocki_, Aug 30 2006

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Last modified January 18 06:34 EST 2019. Contains 319269 sequences. (Running on oeis4.)