A cyclic order on a finite set is an arrangement of its elements around an oriented circle with no element distinguished as first. Equivalently, it is an equivalence class of permutations under cyclic permutations. Reversing the orientation is not identified unless this is stated explicitly.
The number of cyclic orders of 
distinct elements is therefore 
, i.e., the number of circular
permutations.
A cyclic basis ordering applies a cyclic order to the elements of a matroid and requires every window of matroid rank length to be a matroid basis (Kajitani et al. 1988).
Elements Are Independent." Disc. Math. 72,
187-194, 1988.