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Search: id:A122367
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| A122367 |
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Dimension of 3-variable non-commutative harmonics (twisted 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} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j). |
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+0
10
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| 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Essentially identical to A001519
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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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LINKS
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Tanya Khovanova, Recursive Sequences
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FORMULA
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G.f.: (1-q)/(1-3*q+q^2). 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=3 a(n) = 3*a(n-1)-a(n-2) with a(0) = 1, a(1) = 2
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EXAMPLE
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a(1) = 2 because x1-x2, x1-x3 are both of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3
a(2) = 5 because x1 x2 - x3 x2, x1 x3 - x2 x3, x2 x1 - x3 x1, x1 x1 - x2 x1 - x2 x2 + x1 x2, x1 x1 - x3 x1 - x3 x3 + x3 x1 are all lin. ind. and are killed by d_x1+d_x2+d_x3, d_x1 d_x1 + d_x2 d_x2 + d_x3 d_x3 and sum( d_xi d_xj, i,j = 1..3)
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MAPLE
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a:=n->if n=0 then 1; elif n=1 then 2 else 3*a(n-1)-a(n-2); fi;
a:=n->sum(binomial(n+k, 2*k), k=0..n): seq(a(n), n=0..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 02 2007
with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=0), Z, end_blockRL):Q:=subs([a=Union(ZL3), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n), n=1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008
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CROSSREFS
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Cf. A001519, A055105, A055107, A087903, A074664, A008277, A112340, A122368, A122369, A122370, A122371, A122372.
Adjacent sequences: A122364 A122365 A122366 this_sequence A122368 A122369 A122370
Sequence in context: A112844 A027933 A011783 this_sequence A001519 A048575 A099496
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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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