OFFSET
1,1
COMMENTS
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}}.
REFERENCES
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.
LINKS
Richard Bayley, Homepage.
FORMULA
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
EXAMPLE
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
CROSSREFS
KEYWORD
more,nonn
AUTHOR
Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 12 2006
STATUS
approved