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A067370
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The Weight of the Periphery of the alternating group, denoted v(P_N).
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3
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0, 1, 3, 24, 160, 1290, 11046, 106848, 1117152, 12849840, 159089040, 2132602560, 30554297280, 468754715520, 7634862748800, 132058767052800, 2410986506342400, 46443330717235200, 939668036761036800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Sequences A067369, A067370 and A067318 are related. A067318 counts transpositions in the symmetric group, denoted S_n. One can think of the transpositions in S_n as being split between the alternating group A_n and its complement, which we call the periphery and denote P_N. For n>=3, A067319 v(P_N) and A067370 v(A_n) always differ by (n-2)! When n is odd, v(A_n) is larger; when n is even, v(P_N) is larger. This gives new meaning to the name alternating group. The average weight of a permutation in A_n converges with the average weight for a permutation in P_N at infinity.
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FORMULA
| v(P_N)=p(n)=p(n-1)+[(n-1)!/2]*[vbar(A_n-1)+1]*[(n-1)] where vbar(A_n) is the average weight of a permutation in A_n, the alternating group. vbar(A_n-1) is a(n-1)/(n-1)!/2 where a(n) is from the sequence A067369.
a(n) = 1/2*((-1)^n*(n-2)!+n*n!-abs(stirling1(n+1, 2))), n>1. E.g.f.: 1/2*((1+x)*ln(1+x)-x+x/(1-x)^2+log(1-x)/(1-x)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 02 2003
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EXAMPLE
| Let n=4. v(S_n)=46, see A067318. (n-2)!=2!=2. n is even so P_N is larger than A_n. v(P_N)= 23+1=24. v(A_n)=23-1=22, see A067369. Let n=5. v(S_n)=326. (n-2)!=3!=6. n is odd so A_n is larger than P_N. v(P_N)=163-3=160. v(A_n)=163+3=166.
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CROSSREFS
| Cf. A067369 A067318.
Sequence in context: A003443 A119581 A006292 * A094432 A104527 A058038
Adjacent sequences: A067367 A067368 A067369 * A067371 A067372 A067373
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KEYWORD
| easy,nice,nonn
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AUTHOR
| Nick Hann (nickhann(AT)aol.com), Jan 20 2002
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EXTENSIONS
| Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 02 2003
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