1,3

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.

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

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.

Cf. A067369 A067318.

Sequence in context: A003443 A119581 A006292 this_sequence A094432 A104527 A058038

Adjacent sequences: A067367 A067368 A067369 this_sequence A067371 A067372 A067373

easy,nice,nonn

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

Corrected and extended by Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 02 2003