Partial Complements and Transposable Dispersions

Suppose A = {a(i, j)}, for i ≥ 0 and j ≥ 0, is the dispersion of a strictly increasing sequence r = (r(0), r(1), r(2), . . .) of integers, where r(0) = 1 and infinitely many postive integers are not terms of r. Let R be the set of such sequences, and define t on R by tr(n) = a(0, n) for n ≥ 0. Let F be the subset of R consisting of sequences r satisfying ttr = r. The set F is characterized in terms of ordered arrangements of numbers i + jθ. For fixed i ≥ 0, the sequence a(i, j), for j ≥ 1, is the (i + 1)st partial complement of r. Central to the characterization of F is the role of the families of figurate (or polygonal) number sequences and the centered polygonal number sequences. Finally, it is conjectured that for every r in R, the iterates t(2m)r converge to a sequence in F .