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A059867
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Number of irreducible representations of the symmetric group S_n that have odd degree.
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12
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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
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EXAMPLE
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a(3) = 2 because S_3 the degrees of the irreducible representations of S_3 are 1,1,2.
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MATHEMATICA
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a[n_] := 2^Total[Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Noam Katz (noamkj(AT)hotmail.com), Feb 28 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001
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STATUS
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approved
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