The Multiple Permutation Problem and Some Conjectures

In this paper, we proposed an interesting problem that might be classified into enumerative combinatorics. Featuring a distinctive two-fold dependence upon the sequences’ terms, our problem can be really difficult, which calls for novel approaches to work it out for any given pair (m,n). Complete or partial solutions for m = 2, 3 with smaller n’s are listed. Moreover, we have proved the necessary condition for p(m,n) 6= 0 and suggested an elegant asymptotic formula for p(2, n). In addition, several challenging conjectures are provided, together with concise comments.