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A059867 Number of irreducible representations of the symmetric group S_n that have odd degree. 8
1, 2, 2, 4, 4, 8, 8, 8, 8, 16, 16, 32, 32, 64, 64, 16, 16, 32, 32, 64, 64, 128, 128, 128, 128, 256, 256, 512, 512, 1024, 1024, 32, 32, 64, 64, 128, 128, 256, 256, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 4096, 4096 (list; graph; refs; listen; history; text; internal format)
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.

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-Fran├žois Alcover, Jul 21 2018 *)

PROG

(Sage) def A059867(n) : dig = n.digits(2); return prod(2^n for n in xrange(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

Cf. A000120, A029930, A029931 (the bisection), A073642, A089248.

Sequence in context: A120541 A190172 A287293 * A046971 A051754 A108747

Adjacent sequences:  A059864 A059865 A059866 * A059868 A059869 A059870

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

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Last modified December 10 10:38 EST 2018. Contains 318047 sequences. (Running on oeis4.)