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 A133922 a(n) = number of permutations (p(1),p(2),p(3),...p(n)) of (1,2,3,...n) such that p(k) is coprime to p(n+1-k) for k = all positive integers <=n. 1
 1, 2, 2, 16, 16, 192, 192, 6912, 4608, 230400, 230400, 11612160, 11612160, 1199923200, 588349440, 98594979840, 98594979840, 11076328488960, 11076328488960, 2102897147904000, 1076597725593600, 331238941183180800, 331238941183180800, 66325953940291584000, 56326771107377971200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For n = odd integer the middle term of all counted permutations must be 1. From Robert Israel, Sep 12 2016:(Start) Consider the graph with vertices [1,...,n] if n is even, [2,...,n] if n is odd, and edges joining coprime integers. a(n) is A037223(n) times the number of perfect matchings in this graph. If n is even, a(n) = A037223(n)*A009679(n/2). If n is an odd prime, a(n) = a(n-1). (End) LINKS Robert Israel, Table of n, a(n) for n = 1..31 EXAMPLE For n = 6, the permutation (3,2,1,6,4,5) is not counted because p(2)=2 is not coprime to p(5)=4. However, the permutation (3,6,1,4,5,2) is counted because GCD(3,2) = GCD(6,5) = GCD(1,4) = 1. MAPLE M:= proc(A) option remember;     local n, t, i, Ai, Ap, inds, isrt, As;     n:= nops(A);     if n = 0 then return 1 fi;     t:= 0;     for i in A[1] do       inds:= [\$2..i-1, \$i+1..n];       Ai:= subs([1=NULL, i=NULL, seq(inds[i]=i, i=1..n-2)], A[inds]);       isrt:= sort([\$1..n-2], (r, s) -> nops(Ai(r)) <= nops(Ai(s)), output=permutation);       Ai:= subs([seq(isrt[i]=i, i=1..n-2)], Ai[isrt]);       t:= t + procname(Ai);     od;     t; end proc: F:= proc(n) local A;   if n::odd then     if isprime(n) then return procname(n-1) fi;     A:= [seq(select(t -> igcd(t+1, i+1)=1, [\$1..i-1, \$i+1..n-1]), i=1..n-1)];   else     A:= [seq(select(t -> igcd(t, i)=1, [\$1..i-1, \$i+1..n]), i=1..n)]   fi;   M(A)*floor(n/2)!*2^floor(n/2) end proc; seq(F(n), n=1..27); # Robert Israel, Sep 12 2016 CROSSREFS Cf. A009679, A081123, A037223. Sequence in context: A093114 A016740 A193145 * A222954 A240033 A088139 Adjacent sequences:  A133919 A133920 A133921 * A133923 A133924 A133925 KEYWORD hard,nonn AUTHOR Leroy Quet, Jan 07 2008 EXTENSIONS a(6)-a(15) from Sean A. Irvine, May 17 2010 a(16)-a(25) from Robert Israel, Sep 12 2016 STATUS approved

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Last modified May 6 02:35 EDT 2021. Contains 343579 sequences. (Running on oeis4.)