A060840 Number of irreducible representations of symmetric group S_n whose degree is not divisible by 3.

1, 2, 3, 3, 6, 9, 9, 18, 9, 9, 18, 27, 27, 54, 81, 81, 162, 54, 54, 108, 162, 162, 324, 486, 486, 972, 27, 27, 54, 81, 81, 162, 243, 243, 486, 243, 243, 486, 729, 729, 1458, 2187, 2187, 4374, 1458, 1458, 2916, 4374, 4374, 8748, 13122, 13122, 26244, 405, 405, 810

Offset: 1

Author

Noam Katz (noamkj(AT)hotmail.com), May 02 2001

Keywords

• nonn
• easy

Examples

			a(4) = 3 because the degrees for S_4 are 1,1,2,3,3 and by the formula: 4 in base 3 is 11 and a(4) = 1*3
		

References

• I. G. MacDonald, On the degrees of the irreducible representations of symmetric groups, Bull. London Math. Soc. 3 (1971), 189-192

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A059867.

Programs

Mathematica

a[n_] := (id = IntegerDigits[n, 3]; lg = Length[id]; Times @@ Table[ Which[ id[[lg-i]] == 0, 1, id[[lg-i]] == 1, 3^i, True, 3^i*(3^i+3)/2], {i, lg-1, 0, -1}]); Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Apr 30 2013 *)

Sage

def A060840(n) : dig = n.digits(3); return prod([1, 3^m, 3^m*(3^m+3)//2][dig[m]] for m in range(len(dig)))
# Eric M. Schmidt, Apr 30 2013

Formula

If n = sum a_i*3^e[i] in base 3 where a_i is 0, 1, 2 then a(n) = product g(i) where if a(i) = 0 g(i) = 1, if a(i) = 1 g(i) = 3^i, if a(i) = 2 g(i) = 3^i * (3^i + 3) / 2

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 10 2001