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

0, 0, 1, 1, 2, 3, 4, 6, 10, 14, 20, 26, 35, 43, 49, 77, 103, 125, 174, 190, 274, 340, 430, 496, 686, 838, 1026, 1263, 1579, 1832, 2457, 2833, 3631, 4249, 5114, 6111, 7962, 9072, 11015, 12939, 16173, 18304, 23101, 26188, 31822, 37518, 45073, 51403, 63489, 71822

Offset: 0

Author

Pierandrea Formusa, Aug 28 2018

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A060240, A060426, A060437.

Programs

Mathematica

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}]];
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}]]];
a[n_] := a[n] = If[n == 0 || n == 1, 0, Count[Tally[g[n, n, {}]], {, k /; k > 1}] ];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 49}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

SageMath

r=""
lista=[]
lista_rip=[]
rip=0
for i in range(1,35):
        l=Partitions(i)
        for p in l:
            nsc=StandardTableaux(p).cardinality()
            if nsc in lista:
                if nsc not in lista_rip:
                    lista_rip.append(nsc)
                    rip += 1
            else:
                lista.append(nsc)
        r = r+","+str(rip)
        rip=0
        lista=[]
        lista_rip=[]
print(r)

Formula

a(n) = A060437(n) - A060426(n). - Alois P. Heinz, Aug 29 2018

Extensions

a(42)-a(49) from Alois P. Heinz, Aug 29 2018