The Multiple Permutation Problem and Some Conjectures

TL;DR

This paper introduces the multiple permutation problem in combinatorics, providing partial solutions for small cases, a necessary condition, an asymptotic formula for a specific case, and proposing several conjectures.

Contribution

It presents the multiple permutation problem, offers initial solutions for small parameters, and formulates new conjectures along with asymptotic analysis.

Findings

Necessary condition for non-zero solutions established

Asymptotic formula proposed for p(2,n)

Partial solutions for m=2,3 with small n

Abstract

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 $p(m,n) \neq 0$ and suggested an elegant asymptotic formula for $p(2,n)$. In addition, several challenging conjectures are provided, together with concise comments.