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A060426 a(n) is the number of degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n that appear only once.

%I A060426 #21 Sep 23 2024 18:03:39
%S A060426 1,0,1,1,1,1,1,2,1,2,2,3,2,3,4,4,5,5,4,6,8,8,6,7,10,11,11,15,15,16,18,
%T A060426 21,22,23,29,33,31,31,39,43,44,52,51,58,64,71,66,82,88,96,93,103,115,
%U A060426 128,143,150,156,160,173,199,202,202,242,263,269,293,308
%N A060426 a(n) is the number of degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n that appear only once.
%C A060426 Bounded above by A000700(n). - _Eric M. Schmidt_, Apr 29 2013
%e A060426 a(6) = 1 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and the only number that appears once is 16.
%t A060426 h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
%t A060426 g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i < 1, 0, Flatten@ Table[g[n - i*j, i - 1, Join[l, Array[i&, j]]], {j, 0, n/i}]]];
%t A060426 a[n_] := a[n] = If[n == 1, 1, Count[Tally[g[n, n, {}]], {_, 1}] ];
%t A060426 Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* _Jean-François Alcover_, Sep 23 2024, after _Alois P. Heinz_ in A060240 *)
%o A060426 (Sage)
%o A060426 def A060426(n) :
%o A060426     mult = {}
%o A060426     for P in Partitions(n) :
%o A060426         dim = P.dimension()
%o A060426         mult[dim] = mult.get(dim, 0) + 1
%o A060426     return len([m for m in iter(mult) if mult[m]==1])
%o A060426 # _Eric M. Schmidt_, Apr 29 2013
%Y A060426 Cf. A059867, A060368, A060369, A060437, A061569, A089248. [From _M. F. Hasler_, Jun 14 2009]
%K A060426 nonn
%O A060426 1,8
%A A060426 Avi Peretz (njk(AT)netvision.net.il), Apr 05 2001
%E A060426 More terms from _Eric M. Schmidt_, Apr 29 2013