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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). 4
1, 2, 5, 15, 46, 139, 416, 1248, 3744, 11232, 33696, 101088, 303264, 909792, 2729376, 8188128, 24564384, 73693152, 221079456, 663238368, 1989715104, 5969145312, 17907435936, 53722307808, 161166923424, 483500770272, 1450502310816 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

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.

LINKS

Table of n, a(n) for n=0..26.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082 , Canad. J. Math. 60 (2008), no. 2, 266-296.

FORMULA

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

EXAMPLE

a(1) = 2 because x1 - x2, x2 - x3 are killed by d_x1 + d_x2 + d_x3

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,

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

d_x1 d_x2 + d_x1 d_x3 + d_x2 d_x3

MAPLE

coeffs(convert(series(mul(1-q^i, i=1..3)/(1-3*q), q, 20), `+`)-O(q^20), q);

CROSSREFS

Cf. A118264, A122367, A122391, A122393, A122394.

Sequence in context: A065848 A148359 A287582 * A139782 A047086 A071731

Adjacent sequences:  A122389 A122390 A122391 * A122393 A122394 A122395

KEYWORD

nonn,easy

AUTHOR

Mike Zabrocki, Aug 31 2006

STATUS

approved

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Last modified July 1 12:30 EDT 2022. Contains 354973 sequences. (Running on oeis4.)