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A122368
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Dimension of 4-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 4 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).
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5
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1, 3, 11, 42, 162, 627, 2430, 9423, 36549, 141777, 549990, 2133594, 8276985, 32109534, 124565121, 483235875, 1874657763, 7272519066, 28212902154, 109448714619, 424593725526, 1647162628047, 6389978382405, 24789187818585
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| 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
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.
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FORMULA
| o.g.f. (1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3) 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=4
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EXAMPLE
| a(1) = 3 because x1-x2, x2-x3, x3-x4 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4
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MAPLE
| coeffs(convert(series((1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3), q, 30), `+`)-O(q^30), q);
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CROSSREFS
| Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.
Sequence in context: A077830 A106460 A059716 * A032443 A180907 A143464
Adjacent sequences: A122365 A122366 A122367 * A122369 A122370 A122371
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KEYWORD
| nonn
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AUTHOR
| Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006
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