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A124578
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Define p(alpha,2) to be the number of H-conjugacy classes where H is an infant subgroup ( similar to Young subgroups of S_n) of type alpha of the hyperoctahedral group B_n. Then a(n) = sum p(alpha,2) where |alpha| = n and alpha has at most n parts.
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1
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OFFSET
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1,1
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COMMENTS
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p((n,0),2) = A000712. B_n can also be thought of as the signed permutation group. B_3 acts on the alphabet {1,2,3,bar{1}, bar{2}, bar{3}}. An infant subgroup of type (2,1) will be the subgroup which stabilizes the sets {1,bar{1}, 2, bar{2}} and {3,bar{3}}.
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REFERENCES
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Richard Bayley, Relative Character Theory and the Hyperoctahedral Group, Ph.D. thesis, Queen Mary College, University of London, to be published 2007.
Steve Donkin, Invariant functions on Matrices, Math. Proc. Camb. Phil. Soc. 113 (1993) 23-43.
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LINKS
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FORMULA
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Let x = x_1x_2x_3... and x^alpha = x_1^(alpha_1)x_2^(alpha_2)x_3^(alpha_3).... Let Phi = set of all primitive necklaces. If b is a primitive necklace then C(b) = Content(b) = (beta_1, beta_2,beta_3,.....) where beta_i = the number of times i occurs in b. For example if b=[11233] then C(b) = (2,1,2). To generate the p(alpha,2) we do the following. sum_alpha p(alpha,2)x^alpha = prod_(b in Phi) prod_(k = 1)^infinity 1/(1- x^(C(b) times k ))^2 = prod_(b in Phi) prod_(k = 1)^infinity (1+ x^(k times C(b)) + x^(2k times C(b)) + x^(3k times C(b)) + ....)^2
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EXAMPLE
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E.g p((2,1),2) = # H-conjugacy classes of B_3 where H = Inft((2,1)) isom B_2 times B_1 . Then a(3) = p((3),2) + p((2,1),2) + p((2,0,1),2) + p((1,2),2) + p((1,1,1),2)+ p((1,0,2),2)+ p((0,3),2) + p((0,2,1),2) + p((0,1,2),2) + p((0,0,3),2) =10 + 16 + 16 + 16 + 24 + 16 + 10 + 16 + 16 +10 = 150
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CROSSREFS
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KEYWORD
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more,nonn
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AUTHOR
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Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 12 2006
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STATUS
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approved
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