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%I #28 Dec 22 2018 12:17:41
%S 0,1,3,24,160,1290,11046,106848,1117152,12849840,159089040,2132602560,
%T 30554297280,468754715520,7634862748800,132058767052800,
%U 2410986506342400,46443330717235200,939668036761036800,19955747250238464000,443271664862659584000,10290986066890045440000
%N The weight of the periphery of the alternating group, denoted v(P_N).
%C 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, 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 a permutation in A_n converges with the average weight for a permutation in P_N at infinity.
%H Charlie Neder and Muniru A Asiru, <a href="/A067370/b067370.txt">Table of n, a(n) for n = 1..446</a>
%F v(P_N) = p(n) = p(n-1) + floor((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.
%F From _Vladeta Jovovic_, Feb 02 2003: (Start)
%F a(n) = (1/2)*((-1)^n*(n-2)! + n*n! - abs(Stirling1(n+1, 2))), n > 1.
%F E.g.f.: (1/2)*((1+x)*log(1+x) - x + x/(1-x)^2 + log(1-x)/(1-x)). (End)
%e 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.
%p seq(coeff(series(factorial(n)*(1/2)*((1+x)*log(1+x)-x+x/(1-x)^2+log(1-x)/(1-x)),x,n+1), x, n), n = 1 .. 25); # _Muniru A Asiru_, Dec 15 2018
%t a[n_] := (n*n! + (-1)^n*((n-2)! + StirlingS1[n+1, 2]))/2; a[1] = 0; Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, May 23 2012, after _Vladeta Jovovic_ *)
%o (PARI) a(n)={if(n < 2, 0, (1/2)*((-1)^n*(n-2)! + n*n! - abs(stirling(n+1, 2, 1))))} \\ _Andrew Howroyd_, Dec 14 2018
%o (GAP) Concatenation([0],List([2..25],n->(1/2)*((-1)^n*Factorial(n-2)+n*Factorial(n)-AbsInt(Stirling1(n+1,2))))); # _Muniru A Asiru_, Dec 15 2018
%Y Cf. A067369 A067318.
%K easy,nice,nonn
%O 1,3
%A Nick Hann (nickhann(AT)aol.com), Jan 20 2002
%E Corrected and extended by _Vladeta Jovovic_, Feb 02 2003
%E a(20)-a(22) from _Charlie Neder_, Dec 14 2018
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