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Search: id:A122371
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| A122371 |
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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). |
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6
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| 1, 6, 41, 285, 1989, 13901, 97215, 680079, 4758408, 33297267, 233014444, 1630701426, 11412409945, 79870754268, 558989013403, 3912210491549, 27380636068267, 191631324294463, 1341190961828143, 9386756237545989
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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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
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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
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EXAMPLE
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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
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MAPLE
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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)
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CROSSREFS
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Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122368, A122369, A122370, A122372.
Adjacent sequences: A122368 A122369 A122370 this_sequence A122372 A122373 A122374
Sequence in context: A135232 A015551 A049685 this_sequence A083067 A000402 A078009
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KEYWORD
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nonn
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AUTHOR
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Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 30 2006
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