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.

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

Author

Avi Peretz (njk(AT)netvision.net.il), Apr 05 2001

Keywords

• nonn

Comments

Bounded above by A000700(n). - Eric M. Schmidt, Apr 29 2013

Examples

			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.
		

Crossrefs

Cf. A059867, A060368, A060369, A060437, A061569, A089248. [From M. F. Hasler, Jun 14 2009]

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 == 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 *)

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

Extensions

More terms from Eric M. Schmidt, Apr 29 2013