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Search: id:A122370
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| A122370 |
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Dimension of 6-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 6 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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+0
6
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| 1, 5, 29, 172, 1026, 6134, 36712, 219847, 1316963, 7890594, 47282065, 283344410, 1698058817, 10176618298, 60990528210, 365532989831, 2190756912988, 13129979193808, 78692862940748, 471636719623539
(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. (1-10*q+35*q^2-50*q^3+24*q^4)/(1-15*q+81*q^2-192*q^3+189*q^4-53*q^5) 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=6
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EXAMPLE
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a(1) = 5 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4+d_x5+d_x6
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MAPLE
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coeffs(convert(series((1-10*q+35*q^2-50*q^3+24*q^4)/(1-15*q+81*q^2-192*q^3+189*q\ ^4-53*q^5), 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, A122371, A122372.
Sequence in context: A141812 A001653 A141814 this_sequence A088349 A137625 A083066
Adjacent sequences: A122367 A122368 A122369 this_sequence A122371 A122372 A122373
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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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