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A060437
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a(n) is the number of different degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n, i.e., count each degree only once.
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5
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1, 1, 2, 3, 4, 5, 7, 12, 15, 22, 28, 38, 45, 52, 81, 107, 130, 179, 194, 280, 348, 438, 502, 693, 848, 1037, 1274, 1594, 1847, 2473, 2851, 3652, 4271, 5137, 6140, 7995, 9103, 11046, 12978, 16216, 18348, 23153, 26239, 31880, 37582, 45144, 51469, 63571, 71910
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OFFSET
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1,3
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COMMENTS
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The total number of irreducible representations of S_n is the partition function p(n) (sequence A000041) - this is the total number of the degrees counting multiplicities.
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LINKS
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EXAMPLE
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a(6) = 5 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and counting each degree only once only 5 numbers remain: 1,5,9,10,16.
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MAPLE
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with(numtheory):
g:= proc(n) option remember; `if`(n=1, 1,
add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
end:
b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
[seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
end:
a:= n-> nops(map(g, {b(n, n)[]})):
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MATHEMATICA
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g[n_] := g[n] = If[n == 1, 1, Sum[g[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q, FactorInteger[n][[All, 1]]}]]; b[n_, i_] :=b[n, i] = If[n == 0 || i<2, {2^n}, Flatten @ Table[ Map[Function[{p}, p*Prime[i]^j], b[n-i*j, i-1]], {j, 0, n/i}] ]; a[n_] := Length[Union[g /@ b[n, n]]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Avi Peretz (njk(AT)netvision.net.il), Apr 07 2001
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EXTENSIONS
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STATUS
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approved
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