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A124577
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Define p(alpha) to be the number of H-conjugacy classes where H is a Young subgroup of type alpha of the symmetric group S_n. Then a(n) = sum p(alpha) where |alpha| = n and alpha has at most n parts.
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2
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1, 6, 39, 356, 4055, 57786, 983535, 19520264, 441967518, 11235798510, 316719689506, 9800860032876, 330230585628437, 12032866998445818, 471416196117401340, 19758835313514076176, 882185444649249777913
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| p((0,n)) = A000041, p((1,n)) = A000070, p((2,n) = A093695
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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.
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LINKS
| Richard Bayley, Homepage.
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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) we do the following. sum_alpha p(alpha)x^alpha = prod_(b in Phi) prod_(k = 1)^infinity 1/(1- x^(c(b) times k )) = prod_(b in Phi) prod_(k = 1)^infinity (1+ x^(k times C(b)) + x^(2k times C(b)) + x^(3k times C(b)) + ....)
Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 26 2009: (Start)
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k) for n>0.
a(n) = Sum_{k=1..n} A008284(n,k)*n^k, where A008284(n,k) = number of partitions of n in which the greatest part is k, 1<=k<=n. (End)
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EXAMPLE
| E.g p((2,1)) = # H-conjugacy classes of S_3 where H = Yng((2,1)) isom S_2 times S_1 . Then a(3) = p((3)) + p((2,1)) + p((2,0,1)) + p((1,2)) + p((1,1,1))+ p((1,0,2)+ p((0,2,1)) + p((0,1,2)) + p((0,0,3)) = 3+4+4+4+6+4+3+4+4+3 = 39
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PROG
| (GAP)
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-n*x^k +x*O(x^n)), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 26 2009]
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CROSSREFS
| Cf. A124578, A000041, A000070, A093695.
Sequence in context: A058191 A113347 A031972 * A006678 A145709 A034661
Adjacent sequences: A124574 A124575 A124576 * A124578 A124579 A124580
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KEYWORD
| nonn
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AUTHOR
| Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 05 2006
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EXTENSIONS
| Extended with formula by Paul D. Hanna (pauldhanna(AT)juno.com), Nov 26 2009
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