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A122369 Dimension of 5-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 5 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). 7
1, 4, 19, 93, 459, 2273, 11274, 55964, 277924, 1380527, 6858356, 34074280, 169297743, 841173845, 4179517118, 20766807551, 103184684826, 512698227699, 2547469553647, 12657750705603, 62893284231103, 312501512711984 (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.

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

LINKS

Table of n, a(n) for n=0..21.

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

o.g.f. (1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4) 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=5

EXAMPLE

a(1) = 4 because x1-x2, x2-x3, x3-x4, x4-x5 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5

MAPLE

coeffs(convert(series((1-6*q+11*q^2-6*q^3)/(1-10*q+32*q^2-37*q^3+11*q^4), q, 30), `+`)-O(q^30), q)

CROSSREFS

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.

Sequence in context: A288687 A275751 A131552 * A005978 A083065 A137636

Adjacent sequences:  A122366 A122367 A122368 * A122370 A122371 A122372

KEYWORD

nonn

AUTHOR

Mike Zabrocki, Aug 30 2006

STATUS

approved

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Last modified September 22 08:23 EDT 2018. Contains 315270 sequences. (Running on oeis4.)