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A060840
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Number of irreducible representations of symmetric group S_n whose degree is not divisible by 3.
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2
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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
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OFFSET
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1,2
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REFERENCES
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I. G. MacDonald, On the degrees of the irreducible representations of symmetric groups, Bull. London Math. Soc. 3 (1971), 189-192
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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)))
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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), May 02 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), May 10 2001
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STATUS
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approved
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