cp's OEIS Frontend

A122392 Dimension of 3-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).

%I A122392 #10 Dec 10 2013 12:18:31
%S A122392 1,2,5,15,46,139,416,1248,3744,11232,33696,101088,303264,909792,
%T A122392 2729376,8188128,24564384,73693152,221079456,663238368,1989715104,
%U A122392 5969145312,17907435936,53722307808,161166923424,483500770272,1450502310816
%N A122392 Dimension of 3-variable non-commutative harmonics (Hausdorff derivative). The dimension of the space of non-commutative polynomials in 3 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( w ) = sum over all subwords of w deleting xi once).
%D A122392 C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
%D A122392 C. Reutenauer, Free Lie algebras. London Mathematical Society Monographs. New Series, 7. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xviii+269 pp.
%H A122392 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.
%F A122392 G.f.: (1-q)*(1-q^2)*(1-q^3)/(1-3*q) 3^n - 3^(n-1) - 3^(n-2) + 3^(n-4) + 3^(n-5) - 3^(n-6) (for n>5) a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 15, a(4) = 46, a(5) = 139, a(n) = 416*3^(n-6) for n>5
%e A122392 a(1) = 2 because x1 - x2, x2 - x3 are killed by d_x1 + d_x2 + d_x3
%e A122392 a(2) = 5 because x1 x2 - x2 x1, x1 x3 - x3 x1, x2 x3 - x3 x2, 2 x1 x2 - x2 x2 - 2 x1 x3 + x3 x3,
%e A122392 x1 x1 - 2 x2 x1 + 2 x2 x3 - x3 x3 are killed by d_x1 + d_x2 + d_x3, d_x1^2 + d_x2^2 + d_x3^2 and
%e A122392 d_x1 d_x2 + d_x1 d_x3 + d_x2 d_x3
%p A122392 coeffs(convert(series(mul(1-q^i,i=1..3)/(1-3*q),q,20),`+`)-O(q^20),q);
%Y A122392 Cf. A118264, A122367, A122391, A122393, A122394.
%K A122392 nonn,easy
%O A122392 0,2
%A A122392 _Mike Zabrocki_, Aug 31 2006