A067369 Weight of the alternating group (A_n) in transpositions.

0, 0, 4, 22, 166, 1266, 11166, 106128, 1122192, 12809520, 159451920, 2128973760, 30594214080, 468275713920, 7641089769600, 131971588761600, 2412294180710400, 46422407927347200, 940023724189132800, 19949344876532736000, 443393309963068416000, 10288553164881868800000

Offset: 1

Author

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

Keywords

• easy
• nice
• nonn

Comments

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.

Links

• Charlie Neder and Muniru A Asiru, Table of n, a(n) for n = 1..445

Crossrefs

Cf. A067370, A067318.

Programs

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

Maple

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

Mathematica

a[n_] := 1/2*((-1)^(n+1)*(n-2)!+n*n!-Abs[StirlingS1[n+1, 2]]); a[1]=0; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Jan 12 2015, after Vladeta Jovovic *)

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

Formula

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.

From Vladeta Jovovic, Feb 02 2003: (Start)

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)

Extensions

More terms from Vladeta Jovovic, Feb 02 2003

a(20)-a(22) from Charlie Neder, Dec 14 2018