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

1,8

Comments

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

Links

Table of n, a(n) for n=1..67.

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

Adjacent sequences: A060423 A060424 A060425 * A060427 A060428 A060429

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