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Number of degrees of irreducible representations of symmetric group S_n that appear more than once.
0

%I #32 Sep 23 2024 13:38:46

%S 0,0,1,1,2,3,4,6,10,14,20,26,35,43,49,77,103,125,174,190,274,340,430,

%T 496,686,838,1026,1263,1579,1832,2457,2833,3631,4249,5114,6111,7962,

%U 9072,11015,12939,16173,18304,23101,26188,31822,37518,45073,51403,63489,71822

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

%F a(n) = A060437(n) - A060426(n). - _Alois P. Heinz_, Aug 29 2018

%e 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 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 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 a[n_] := a[n] = If[n == 0 || n == 1, 0, Count[Tally[g[n, n, {}]], {_, k_ /; k > 1}] ];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 49}] (* _Jean-François Alcover_, Sep 23 2024, after _Alois P. Heinz_ in A060240 *)

%o (SageMath)

%o r=""

%o lista=[]

%o lista_rip=[]

%o rip=0

%o for i in range(1,35):

%o l=Partitions(i)

%o for p in l:

%o nsc=StandardTableaux(p).cardinality()

%o if nsc in lista:

%o if nsc not in lista_rip:

%o lista_rip.append(nsc)

%o rip += 1

%o else:

%o lista.append(nsc)

%o r = r+","+str(rip)

%o rip=0

%o lista=[]

%o lista_rip=[]

%o print(r)

%Y Cf. A060240, A060426, A060437.

%K nonn

%O 0,5

%A _Pierandrea Formusa_, Aug 28 2018

%E a(42)-a(49) from _Alois P. Heinz_, Aug 29 2018