OFFSET
1,2
COMMENTS
Ayyer et al. (2016, 2016) obtain this sequence (which they call "odd partitions") as the number of partitions of n such that the dimension of the corresponding irreducible representation of S_n is odd.
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 1..1000
Arvind Ayyer, Amritanshu Prasad, Steven Spallone, Odd partitions in Young's lattice, arXiv:1601.01776 [math.CO], 2016.
Arvind Ayyer, A. Prasad, S. Spallone, Representations of symmetric groups with non-trivial determinant, arXiv preprint arXiv:1604.08837 [math.RT], 2016. See Eq. (14).
I. G. Macdonald, On the degrees of the irreducible representations of symmetric groups, Bulletin of the London Mathematical Society, 3(2):189-192, 1971.
John McKay, Irreducible representations of odd degree, Journal of Algebra 20, 1972 pages 416-418.
Igor Pak, Greta Panova, Bounds on Kronecker coefficients via contingency tables, Linear Algebra and its Applications (2020), Vol. 602, 157-178.
FORMULA
If n = sum 2^e[i] in binary, then the number of odd degree irreducible complex representations of S_n is 2^sum e[i]. In words: write n in binary and take the product of the powers of 2 that appear.
G.f.: prod(k>=0, 1 + 2^k * x^2^k). a(n) = 2^A073642(n). - Ralf Stephan, Jun 02 2003
a(1)=1, a(2n) = 2^e1(n)*a(n), a(2n+1) = a(2n), where e1(n) = A000120(n). - Ralf Stephan, Jun 19 2003
EXAMPLE
a(3) = 2 because S_3 the degrees of the irreducible representations of S_3 are 1,1,2.
MATHEMATICA
a[n_] := 2^Total[Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1];
Array[a, 60] (* Jean-François Alcover, Jul 21 2018 *)
PROG
(Sage) def A059867(n) : dig = n.digits(2); return prod(2^n for n in range(len(dig)) if dig[n]==1) # Eric M. Schmidt, Apr 27 2013
(PARI) A059867(n)={my(d=binary(n)); prod(k=1, #d, if(d[#d+1-k], 2^(k-1), 1)); } \\ Joerg Arndt, Apr 29 2013
(PARI) a(n) = {my(b = Vecrev(binary(n))); 2^sum(k=1, #b, (k-1)*b[k]); } \\ Michel Marcus, Jan 11 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Noam Katz (noamkj(AT)hotmail.com), Feb 28 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
STATUS
approved