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A067369
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Weight of the alternating group (A_n) in transpositions.
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3
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0, 0, 4, 22, 166, 1266, 11166, 106128, 1122192, 12809520, 159451920, 2128973760, 30594214080, 468275713920, 7641089769600, 131971588761600, 2412294180710400, 46422407927347200, 940023724189132800, 19949344876532736000, 443393309963068416000, 10288553164881868800000
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OFFSET
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1,3
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COMMENTS
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Sequences A067369, A067370 and A067318 are related: A067318 = A067369 + A067370. 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, A067369 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 permutation in A_n converges with the average weight for a permutation in P_N at infinity.
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LINKS
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FORMULA
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a(n) = a(n-1) + [(n-1)!/2]*[vbar(P_N-1)+1]*[n-1)] where vbar(P_N) is the average weight of a permutation in P_N, the periphery of A_n. vbar(P_N-1) is p(n-1)/(n-1)!2 where p(n) is from sequence A067370.
a(n) = (1/2)*((-1)^(n+1)*(n-2)! + n*n! - abs(Stirling1(n+1, 2))), n > 1.
E.g.f.: (1/2)*(-(1+x)*log(1+x) + x + x/(1-x)^2 + log(1-x)/(1-x) + 2). (End)
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MAPLE
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seq(coeff(series(factorial(n)*(1/2)*(-(1+x)*log(1+x)+x+x/(1-x)^2+log(1-x)/(1-x)+2), x, n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Dec 15 2018
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MATHEMATICA
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PROG
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(PARI) a(n)={if(n < 2, 0, 1/2*((-1)^(n+1)*(n-2)!+n*n!-abs(stirling(n+1, 2, 1))))} \\ Andrew Howroyd, Dec 14 2018
(GAP) Concatenation([0], List([2..25], n->(1/2)*((-1)^(n+1)*Factorial(n-2)+n*Factorial(n)-AbsInt(Stirling1(n+1, 2))))); # Muniru A Asiru, Dec 15 2018
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Nick Hann (nickhann(AT)aol.com), Jan 20 2002
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EXTENSIONS
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STATUS
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approved
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