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Search: id:A122372
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| A122372 |
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Dimension of 8-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 8 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, 7, 55, 438, 3498, 27962, 223604, 1788406, 14305102, 114429193, 915366442, 7322521512, 58577537621, 468602617723, 3748697751384, 29988696932490, 239903055854075, 1919175464438065, 15353030007717639, 122821355074655309
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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coeffs(convert(series((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/(1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7),q,20),`+`)-O(q^20),q)
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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. (1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/(1-28*q+316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7) 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=8
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EXAMPLE
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A122371 a(1) = 7 because x1-x2, x2-x3, x3-x4, x4-x5, x5-x6, x6-x7, x7-x8 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((1-21*q+175*q^2-735*q^3+1624*q^4-1764*q^5+720*q^6)/(1-28*q\ +316*q^2-1845*q^3+5925*q^4-10190*q^5+8249*q^6-2119*q^7), 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, A122371.
Sequence in context: A069404 A015564 A070997 this_sequence A083068 A097189 A049028
Adjacent sequences: A122369 A122370 A122371 this_sequence A122373 A122374 A122375
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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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