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A060426
a(n) is the number of degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n that appear only once.
2
1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 4, 4, 5, 5, 4, 6, 8, 8, 6, 7, 10, 11, 11, 15, 15, 16, 18, 21, 22, 23, 29, 33, 31, 31, 39, 43, 44, 52, 51, 58, 64, 71, 66, 82, 88, 96, 93, 103, 115, 128, 143, 150, 156, 160, 173, 199, 202, 202, 242, 263, 269, 293, 308
OFFSET
1,8
COMMENTS
Bounded above by A000700(n). - Eric M. Schmidt, Apr 29 2013
EXAMPLE
a(6) = 1 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and the only number that appears once is 16.
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 == 1, 1, Count[Tally[g[n, n, {}]], {_, 1}] ];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)
PROG
(Sage)
def A060426(n) :
mult = {}
for P in Partitions(n) :
dim = P.dimension()
mult[dim] = mult.get(dim, 0) + 1
return len([m for m in iter(mult) if mult[m]==1])
# Eric M. Schmidt, Apr 29 2013
CROSSREFS
Cf. A059867, A060368, A060369, A060437, A061569, A089248. [From M. F. Hasler, Jun 14 2009]
Sequence in context: A103273 A289120 A025066 * A260412 A360566 A283451
KEYWORD
nonn
AUTHOR
Avi Peretz (njk(AT)netvision.net.il), Apr 05 2001
EXTENSIONS
More terms from Eric M. Schmidt, Apr 29 2013
STATUS
approved