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A318558 Number of degrees of irreducible representations of symmetric group S_n that appear more than once.

%I A318558 #32 Sep 23 2024 13:38:46
%S A318558 0,0,1,1,2,3,4,6,10,14,20,26,35,43,49,77,103,125,174,190,274,340,430,
%T A318558 496,686,838,1026,1263,1579,1832,2457,2833,3631,4249,5114,6111,7962,
%U A318558 9072,11015,12939,16173,18304,23101,26188,31822,37518,45073,51403,63489,71822
%N A318558 Number of degrees of irreducible representations of symmetric group S_n that appear more than once.
%F A318558 a(n) = A060437(n) - A060426(n). - _Alois P. Heinz_, Aug 29 2018
%e A318558 Number 4 has the following partitions: a) [4], b) [3, 1], c) [2, 2], d) [2, 1, 1], e) [1, 1, 1, 1]. For partition a the cardinality of standard Young tableaux is 1, for b 3, for c 2, for d 3 and for e 1, so multiple cardinalities are 1 and 3: two multiple cardinalities, i.e., 4th sequence element is 2.
%t A318558 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 A318558 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 A318558 a[n_] := a[n] = If[n == 0 || n == 1, 0, Count[Tally[g[n, n, {}]], {_, k_ /; k > 1}] ];
%t A318558 Table[Print[n, " ", a[n]]; a[n], {n, 0, 49}] (* _Jean-François Alcover_, Sep 23 2024, after _Alois P. Heinz_ in A060240 *)
%o A318558 (SageMath)
%o A318558 r=""
%o A318558 lista=[]
%o A318558 lista_rip=[]
%o A318558 rip=0
%o A318558 for i in range(1,35):
%o A318558         l=Partitions(i)
%o A318558         for p in l:
%o A318558             nsc=StandardTableaux(p).cardinality()
%o A318558             if nsc in lista:
%o A318558                 if nsc not in lista_rip:
%o A318558                     lista_rip.append(nsc)
%o A318558                     rip += 1
%o A318558             else:
%o A318558                 lista.append(nsc)
%o A318558         r = r+","+str(rip)
%o A318558         rip=0
%o A318558         lista=[]
%o A318558         lista_rip=[]
%o A318558 print(r)
%Y A318558 Cf. A060240, A060426, A060437.
%K A318558 nonn
%O A318558 0,5
%A A318558 _Pierandrea Formusa_, Aug 28 2018
%E A318558 a(42)-a(49) from _Alois P. Heinz_, Aug 29 2018