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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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16
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1, 1, 6, 39, 356, 4055, 57786, 983535, 19520264, 441967518, 11235798510, 316719689506, 9800860032876, 330230585628437, 12032866998445818, 471416196117401340, 19758835313514076176, 882185444649249777913, 41797472220815112375966, 2094455101139881670407954
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OFFSET
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0,3
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COMMENTS
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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
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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) 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)) + ....)
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
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EXAMPLE
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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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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n)=polcoeff(1/prod(k=1, n, 1-n*x^k +x*O(x^n)), n)} \\ Paul D. Hanna, Nov 26 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 05 2006
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EXTENSIONS
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STATUS
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approved
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