The problem of irregular perfect systems of sets of iterated differences

Fors2, the set of iterated differences associated with the prescribed integersa(s, j), 1js, is the set {a(i, j): 1jis} wherea(i–1,j)=|a(i, j)–a(i, j+1)|, general problem raised by work of Kreweras and Loeb concerns the existence of partitions of runs of consecutive integers into full sets of iterated differences. In the regular case, where all the sets of iterated differences have the same valencys, it is known that such partitions do not exist at least fors>8. We find here that the problem is more challenging in the case where the sets have different valencies.