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A124577 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. 16
1, 1, 6, 39, 356, 4055, 57786, 983535, 19520264, 441967518, 11235798510, 316719689506, 9800860032876, 330230585628437, 12032866998445818, 471416196117401340, 19758835313514076176, 882185444649249777913, 41797472220815112375966, 2094455101139881670407954 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
p((0,n)) = A000041, p((1,n)) = A000070, p((2,n) = A093695;
Also main diagonal of A209664. - Wouter Meeussen, Mar 11 2012
Number of partitions of n into n sorts of parts. a(2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b]. - Alois P. Heinz, Sep 08 2014
LINKS
Richard Bayley, Homepage.
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.
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)) + ....)
From Paul D. Hanna, 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)
a(n) ~ n^n * (1 + 1/n + 2/n^2 + 3/n^3 + 5/n^4 + 7/n^5 + 11/n^6 + 15/n^7 + 22/n^8 + 30/n^9 + 42/n^10), where the coefficients are A000041(k)/n^k. - Vaclav Kotesovec, Mar 19 2015
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.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
end:
a:= n-> b(n$3):
seq(a(n), n=0..20); # Alois P. Heinz, Sep 08 2014
MATHEMATICA
p[n_Integer, v_] := Sum[Subscript[x, j]^n, {j, v}];
p[par_List, v_] := Times @@ (p[#, v] & /@ par);
Tr /@ Table[(p[#, l] & /@ IntegerPartitions[l]) /. Subscript[x, _] -> 1, {l, 19}] (* Wouter Meeussen, Mar 11 2012 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + If[i > n, 0, k*b[n - i, i, k]]]]; a[n_] := b[n, n, n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
PROG
(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-n*x^k +x*O(x^n)), n)} \\ Paul D. Hanna, Nov 26 2009
CROSSREFS
Main diagonal of A246935.
Sequence in context: A031972 A356439 A308861 * A352839 A006678 A252761
KEYWORD
nonn
AUTHOR
Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 05 2006
EXTENSIONS
Extended with formula by Paul D. Hanna, Nov 26 2009
a(0) inserted and more terms from Alois P. Heinz, Sep 08 2014
STATUS
approved

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Last modified March 29 09:59 EDT 2024. Contains 371268 sequences. (Running on oeis4.)