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A060369 a(n) is the maximum number of occurrences of a degree in the sequence of the degrees of the irreducible representations of the symmetric group S_n.

%I A060369 #13 Sep 23 2024 11:32:06
%S A060369 1,2,2,2,2,4,4,2,3,2,4,4,6,6,6,4,8,6,10,6,8,8,12,8,12,12,10,12,22,14,
%T A060369 12,12,16,18,30,14,20,26,16,20,22,20,26,25,24,24,32,16,32,30,26,24,32,
%U A060369 32,40,32,34,32,32,34,44,30,44,36,52,34,54,38,56,50
%N A060369 a(n) is the maximum number of occurrences of a degree in the sequence of the degrees of the irreducible representations of the symmetric group S_n.
%e A060369 a(6) = 4 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and the number 5 appears 4 times.
%t A060369 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 A060369 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 A060369 a[n_] := a[n] = If[n == 1, 1, MaximalBy[Tally[g[n, n, {}]], Last][[1, 2]]];
%t A060369 Table[Print[n, " ", a[n]];
%t A060369 a[n], {n, 1, 50}] (* _Jean-François Alcover_, Sep 23 2024, after _Alois P. Heinz_ in A060240 *)
%o A060369 (Sage)
%o A060369 def A060369(n) :
%o A060369     mult = {}
%o A060369     for P in Partitions(n):
%o A060369         dim = P.dimension()
%o A060369         mult[dim] = mult.get(dim, 0) + 1
%o A060369     return max(mult.values())
%o A060369 # _Eric M. Schmidt_, May 01 2013
%K A060369 nonn
%O A060369 1,2
%A A060369 Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001
%E A060369 More terms from _Eric M. Schmidt_, May 01 2013