1,3

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, 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 permutation in A_n converges with the average weight for a permutation in P_N at infinity.

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)*ln(1+x)+x+x/(1-x)^2+log(1-x)/(1-x)+2). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 02 2003

Cf. A067370, A067318.

Adjacent sequences: A067366 A067367 A067368 this_sequence A067370 A067371 A067372

Sequence in context: A089464 A111343 A121397 this_sequence A113351 A001827 A125863

easy,nice,nonn

Nick Hann (nickhann(AT)aol.com), Jan 20 2002

More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 02 2003