Regular perfect systems of sets of iterated differences (Q1378293)

Given a vector \(a=\langle a_1,\dots,a_s\rangle\) of integers construe the vector \(b=\langle b_1,\dots,b_{s-1}\rangle\) setting \(b_i=|a_i-a_{i+1}|\). By iteration we obtain \(\bigtriangledown (a)\). Starting by a collection of \(m\) vectors of size \(s\) we have a perfect system of iterated differences of threshold \(c\) (called an \([m,s,c]\)-system) if the set of integers involved is the set of different consecutive integers starting with \(c\). Many necessary conditions to have \([m,s,c]\)-systems are given, e.g. \(2\leq s\leq 9\); if \(s=9\) then \(3m\geq 13(2c-1)\); if \(m=2c-1\) then \(s\leq 5\). Selected examples are given. The set of numbers \(\sum_{k=i}^ja_k\) is called the difference set \(D(a)\) of the vector \(a\). It is called full if it contains distinct integers. The difference sets obtained by the vectors of an \([m,s,c]\)-system are very linked with the system.