login
A318558
Number of degrees of irreducible representations of symmetric group S_n that appear more than once.
0
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
0,5
FORMULA
a(n) = A060437(n) - A060426(n). - Alois P. Heinz, Aug 29 2018
EXAMPLE
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.
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 *)
PROG
(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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Pierandrea Formusa, Aug 28 2018
EXTENSIONS
a(42)-a(49) from Alois P. Heinz, Aug 29 2018
STATUS
approved