

A228772


Number of undirected circular permutations i_0,i_1,...,i_{n1} of 0,1,...,n1 such that i_0+i_1+i_2, i_1+i_2+i_3, ..., i_{n3}+i_{n2}+i_{n1}, i_{n2}+i_{n1}+i_0, i_{n1}+i_0+i_1 are pairwise distinct modulo n.


2




OFFSET

3,2


COMMENTS

Note that if n > 3 is not a multiple of 3 then a(n) > 0 since the natural circular permutation (0,1,2,...,n1) 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_{n2}+a_{n1}+a_n, a_{n1}+a_n+a_1, a_n+a_1+a_2 are pairwise distinct.
On Sep 13 2013, the author proved the conjecture for any torsionfree abelian group G.


REFERENCES

ZhiWei Sun, An additive theorem and restricted sumsets, Math. Res. Lett. 15(2008), 12631276.


LINKS

Table of n, a(n) for n=3..9.
ZhiWei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679.


EXAMPLE

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).


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!}]


CROSSREFS

Cf. A228626, A228766, A185645, A228728.
Sequence in context: A151429 A151475 A105525 * A165714 A090883 A100645
Adjacent sequences: A228769 A228770 A228771 * A228773 A228774 A228775


KEYWORD

nonn,more,hard


AUTHOR

ZhiWei Sun, Sep 03 2013


STATUS

approved



