Search: id:A122371
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%I A122371
%S A122371 1,6,41,285,1989,13901,97215,680079,4758408,33297267,233014444,
%T A122371 1630701426,11412409945,79870754268,558989013403,3912210491549,
%U A122371 27380636068267,191631324294463,1341190961828143,9386756237545989
%N 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).
%D A122371 N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants 
               of the Symmetric Group in Noncommuting Variables, to appear Canad. 
               J. Math., arXiv:math.CO/0502082
%D A122371 C. Chevalley, Invariants of finite groups generated by reflections, Amer. 
               J. Math. 77 (1955), 778-782.
%D A122371 M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. 
               J. 2 (1936), 626-637.
%F A122371 o.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 A122371 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 A122371 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)
%Y A122371 Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, 
               A122369, A122370, A122372.
%Y A122371 Sequence in context: A135232 A015551 A049685 this_sequence A083067 A000402 
               A152107
%Y A122371 Adjacent sequences: A122368 A122369 A122370 this_sequence A122372 A122373 
               A122374
%K A122371 nonn
%O A122371 0,2
%A A122371 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006

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