A089248 a(n) is the sum of the odd degrees of the irreducible representations of the symmetric group S_n.

1, 2, 2, 8, 12, 40, 144, 128, 644, 3504, 7000, 48224, 130992, 861792, 3257600, 32768, 425988, 5833312, 27621672, 415526656, 1987852432, 17674429440, 157807273408, 265515959680, 2848581615344, 30980959604096, 114059874705248, 1365388896050048, 6215927122198944

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Dec 11 2003

Keywords

• nonn

Comments

a(n) is divisible by 4 for n >= 4. - Eric M. Schmidt, Apr 28 2013

References

• John McKay, Irreducible representations of odd degree, Journal of Algebra 20, 1972 pages 416-418.

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..200
• Eric M. Schmidt, Sage code to compute this sequence

Crossrefs

Cf. A000085, A059867.

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, Select[g[n, n, {}], OddQ] // Total];
Table[Print[n, " ", a[n]];
a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

Sage

# Simple but inefficient; see links for faster code
def A089248(n) :
    res = 0
    for P in Partitions(n) :
        deg = P.dimension()
        if is_odd(deg) : res += deg
    return res
# Eric M. Schmidt, Apr 28 2013

Formula

a(2^n) = 2^(2^n - 1). - Eric M. Schmidt, Apr 28 2013

Extensions

More terms from Eric M. Schmidt, Apr 28 2013