1,1

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}}.

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.

Richard Bayley, Homepage.

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

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

Cf. A124577, A000712.

Sequence in context: A003768 A024915 A103885 this_sequence A085510 A012391 A012387

Adjacent sequences: A124575 A124576 A124577 this_sequence A124579 A124580 A124581

more,nonn

Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 12 2006