A228772 Number of undirected circular permutations i_0,i_1,...,i_{n-1} of 0,1,...,n-1 such that i_0+i_1+i_2, i_1+i_2+i_3, ..., i_{n-3}+i_{n-2}+i_{n-1}, i_{n-2}+i_{n-1}+i_0, i_{n-1}+i_0+i_1 are pairwise distinct modulo n.

0, 3, 2, 24, 24, 392, 513, 4080, 8090, 96816, 238296, 2023896, 7325520, 63277376, 277838352, 2185076682, 12898278126

Offset: 3

Author

Zhi-Wei Sun, Sep 03 2013

Keywords

• nonn
• more
• hard

Comments

Note that if n > 3 is not a multiple of 3 then a(n) > 0 since the natural circular permutation (0,1,2,...,n-1) meets the requirement.

Conjecture: Let G be an additive abelian group. If G is cyclic or G contains no involution, then for any finite subset A of G with |A| = n > 3, there is a numbering a_1,...,a_n of the elements of A such that the n sums a_1+a+2+a_3, a_2+a_3+a_4, ..., a_{n-2}+a_{n-1}+a_n, a_{n-1}+a_n+a_1, a_n+a_1+a_2 are pairwise distinct.

On Sep 13 2013, the author proved the conjecture for any torsion-free abelian group G.

Examples

			a(4) = 3 due to the circular permutations (0,1,2,3), (0,1,3,2) and (0,2,1,3).
a(5) = 2 due to the circular permutations (0,1,2,3,4) and(0,2,4,1,3).
a(6) > 0 due to the circular permutation (0,1,2,4,5,3).
a(9) > 0 due to the circular permutation (0,1,2,3,8,5,6,7,4).
		

Links

• Zhi-Wei Sun, An additive theorem and restricted sumsets, Math. Res. Lett. 15(2008), 1263-1276.
• Zhi-Wei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014.

Crossrefs

Cf. A228626, A228766, A185645, A228728.

Programs

Mathematica

(* A program to compute required circular permutations for n = 9. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction; for example, (0,4,7,6,5,8,3,2,1) is identical to (0,1,2,3,8,5,6,7,4) if we ignore direction.*)
V[i_]:=Part[Permutations[{1,2,3,4,5,6,7,8}],i]
m=0
Do[If[Length[Union[Table[Mod[If[j==0,0,Part[V[i],j]]+If[j<8,Part[V[i],j+1],0]+If[j<7,Part[V[i],j+2],If[j==7,0,Part[V[i],1]]],9],{j,0,8}]]]<9,Goto[aa]];
m=m+1;Print[m,":"," ",0," ",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]," ",Part[V[i],8]];Label[aa];Continue,{i,1,8!}]

Extensions

a(10)-a(18) from Bert Dobbelaere, Sep 08 2019

a(19) from Robin Visser, Sep 24 2023