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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.

%I A124577 #44 Sep 17 2024 20:51:03
%S A124577 1,1,6,39,356,4055,57786,983535,19520264,441967518,11235798510,
%T A124577 316719689506,9800860032876,330230585628437,12032866998445818,
%U A124577 471416196117401340,19758835313514076176,882185444649249777913,41797472220815112375966,2094455101139881670407954
%N 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.
%C A124577 p((0,n)) = A000041, p((1,n)) = A000070, p((2,n)) = A093695;
%C A124577 Also main diagonal of A209664. - _Wouter Meeussen_, Mar 11 2012
%C A124577 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
%H A124577 Alois P. Heinz, <a href="/A124577/b124577.txt">Table of n, a(n) for n = 0..250</a>
%H A124577 Richard Bayley, <a href="http://www.maths.qmul.ac.uk/~rtb">Homepage</a>.
%H A124577 Richard Bayley, <a href="http://www.maths.qmul.ac.uk/~rtb/mathpage/Richard%20Bayley&#39;s%20Homepage_files/relhyp2.pdf">Relative Character Theory and the Hyperoctahedral Group</a>, Ph.D. thesis, Queen Mary College, University of London, to be published 2007.
%H A124577 Steve Donkin, <a href="http://dx.doi.org/10.1017/S0305004100075757">Invariant functions on Matrices</a>, Math. Proc. Camb. Phil. Soc. 113 (1993) 23-43.
%H A124577 Wikipedia, <a href="http://en.wikipedia.org/wiki/Symmetric_polynomials">Symmetric Polynomials</a>
%F A124577 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)) + ....)
%F A124577 From _Paul D. Hanna_, Nov 26 2009: (Start)
%F A124577 a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k) for n>0.
%F A124577 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)
%F A124577 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
%e A124577 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.
%p A124577 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A124577       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
%p A124577     end:
%p A124577 a:= n-> b(n$3):
%p A124577 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 08 2014
%t A124577 p[n_Integer, v_] := Sum[Subscript[x, j]^n, {j, v}];
%t A124577 p[par_List, v_] := Times @@ (p[#, v] & /@ par);
%t A124577 Tr /@ Table[(p[#, l] & /@ IntegerPartitions[l]) /. Subscript[x, _] -> 1, {l, 19}] (* _Wouter Meeussen_, Mar 11 2012 *)
%t A124577 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_ *)
%o A124577 (PARI) {a(n)=polcoeff(1/prod(k=1,n,1-n*x^k +x*O(x^n)),n)} \\ _Paul D. Hanna_, Nov 26 2009
%Y A124577 Cf. A124578, A000041, A000070, A093695, A209664-A209673, A291698.
%Y A124577 Main diagonal of A246935.
%K A124577 nonn
%O A124577 0,3
%A A124577 Richard Bayley (r.t.bayley(AT)qmul.ac.uk), Nov 05 2006
%E A124577 Extended with formula by _Paul D. Hanna_, Nov 26 2009
%E A124577 a(0) inserted and more terms from _Alois P. Heinz_, Sep 08 2014