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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. 15
 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 Alois P. Heinz, Table of n, a(n) for n = 0..250 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. Wikipedia, Symmetric Polynomials 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 Cf. A124578, A000041, A000070, A093695, A209664-A209673, A291698. Main diagonal of A246935. Sequence in context: A246571 A031972 A308861 * A006678 A252761 A145709 Adjacent sequences:  A124574 A124575 A124576 * A124578 A124579 A124580 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 September 21 10:03 EDT 2020. Contains 337268 sequences. (Running on oeis4.)