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A322069 Number of permutations f of {1,...,n} with f(1) < f(n) such that Sum_{k=1..n-1} 1/(f(k)*f(k+1)) = 1.

%I A322069 #18 Aug 20 2022 08:50:36
%S A322069 0,0,0,0,0,3,1,7,13,83,84,540,480,5488,48922
%N A322069 Number of permutations f of {1,...,n} with f(1) < f(n) such that Sum_{k=1..n-1} 1/(f(k)*f(k+1)) = 1.
%C A322069 Conjecture 1: a(n) > 0 for all n > 5. In other words, for each n = 6,7,... we have Sum_{k=1..n-1} 1/(f(k)*f(k+1)) = 1 for some permutation f in the symmetric group S_n.
%C A322069 Conjecture 2: For any integer n > 6, there is an undirected circular permutation g different from the circular permutation (1,2,...,n) such that 1/(g(1)*g(2)) + 1/(g(2)*g(3)) + ... + 1/(g(n-1)*g(n)) + 1/(g(n)*g(1)) = 1.
%C A322069 We have verified both conjectures for n up to 11. For Conjecture 2 with n = 7, we may take (g(1),...,g(7)) = (3,2,1,6,5,4,7) since 1/(3*2) + 1/(2*1) + 1/(1*6) + 1/(6*5) + 1/(5*4) + 1/(4*7) + 1/(7*3) = 1.
%C A322069 See also A322070 for a similar conjecture.
%H A322069 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1811.10503">On permutations of {1, ..., n} and related topics</a>, arXiv:1811.10503 [math.CO], 2018.
%e A322069 a(7) = 1, and for the permutation (2,1,3,7,4,5,6) of {1,...,7} we have 1/(2*1) + 1/(1*3) + 1/(3*7) + 1/(7*4) + 1/(4*5) + 1/(5*6) = 1.
%t A322069 V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]];
%t A322069 Do[r=0;Do[If[Part[V[n],k][[1]]>=Part[V[n],k][[n]]||Sum[1/(Part[V[n],k][[i]]*Part[V[n],k][[i+1]]),{i,1,n-1}]!=1,Goto[aa]];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]
%Y A322069 Cf. A322070.
%K A322069 nonn,more
%O A322069 1,6
%A A322069 _Zhi-Wei Sun_, Nov 25 2018
%E A322069 a(12)-a(15) from _Hugo Pfoertner_, Aug 20 2022