|
%I #47 Aug 23 2023 08:44:04
%S 1,1,0,1,1,2,8,20,18,166,397,2788,5448,78102,149562,2576896,6003432,
%T 91012592,257246112,5272355344,12450552690
%N Number of permutations i_1, ..., i_n of 1, ..., n with i_1 = 1 and i_n = n such that i_1+2*i_2, i_2+2*i_3, ..., i_{n-1}+2*i_n, i_n+2*i_1 are pairwise distinct modulo n.
%C If n is not divisible by 3 then a(n) > 0 since the identical permutation of 1 ,..., n works for the purpose. If n is even, then a(n) > 0 since the permutation 1, 2, n-1, 4, n-3, 6, n-5, ..., n-2, 3, n meets the requirement. We guess that a(n) > 0 in the remaining case n = 6q+3 with q > 0. If n = 2k+1 == 3 (mod 6) with n > 3, then, for the permutation (i_1,...,i_n) = (1,2k,k,2k-1,k-1,2k-2,...,3,k+2,2,k+1,2k+1), all the n sums i_1+2*i_2, i_2+2*i_3, ..., i_{n-1}+2*i_n, i_n+2*i_1 are pairwise distinct (but they are not pairwise incongruent modulo n = 2k+1 when n > 9).
%C Zhi-Wei Sun also made the following general conjecture:
%C (i) (Weak version) Let a_1,...,a_n be n distinct elements of an additive abelian group G. Then, there is a permutation b_1,...,b_n of a_1,...,a_n such that a_1+2*b_1, a_2+2*b_2, ..., a_n+2*b_n are pairwise distinct. (The author has proved this for n up to 4 in any abelian group G.)
%C (ii) (Strong version) Let A be any subset of an additive abelian group G with |A| = n > 4. Then there is a numbering a_1, ..., a_n of all the elements of A such that a_1+2*a_2, a_2+2*a_3, ..., a_{n-1}+2*a_n, a_n+2*a_1 are pairwise distinct. (The author has proved this for any torsion-free abelian group G.)
%C Recall that a conjecture of Snevily proved by Arsovski states that for any two n-subsets A and B of an additive abelian group of odd order there is a numbering a_1,...,a_n of all the elements of A and a numbering b_1, ..., b_n of all the elements of B such that the n sums a_1+b_1, ..., a_n+b_n are pairwise distinct.
%H B. Arsovski, <a href="https://doi.org/10.1007/s11856-011-0040-6">A proof of Snevily's conjecture</a>, Israel J. Math. 182(2011), 505-508.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1309.1679">Some new problems in additive combinatorics</a>, preprint, arXiv:1309.1679 [math.NT], 2013-2014.
%e a(6) = 2 due to the permutations 1,2,5,4,3,6 and 1,4,3,2,5,6.
%e a(9) > 0 due to the permutation 1,2,3,5,8,4,7,6,9.
%e a(12) > 0 due to the permutation 1,2,3,4,6,8,5,11,10,7,9,12.
%t (* A program to compute desired permutations for n = 9. *)
%t V[i_]:=Part[Permutations[{2,3,4,5,6,7,8}],i]
%t m=0
%t Do[If[Length[Union[{2},Table[Mod[If[j==0,1,Part[V[i],j]]+2*If[j<7,Part[V[i],j+1],9],9],{j,0,7}]]]<9,Goto[aa]];
%t m=m+1;Print[m,":"," ",1," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]," ",9];Label[aa];Continue,{i,1,7!}]
%Y Cf. A228772, A228766.
%K nonn,more,hard
%O 1,6
%A _Zhi-Wei Sun_, Sep 20 2013
%E a(12)-a(21) from _Robin Visser_, Aug 21 2023
|
