Partial Complements and Transposable Dispersions

Suppose $A=\{a(i,j)\},$ for $i\geq 0$ and $j\geq 0,$ is the dispersion of a strictly increasing sequence $r=(r(0),r(1),r(2),\ldots )$ 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 $n\geq 0.$ 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 $i+j\theta$ . For fixed $i\geq 0,$ the sequence a(i,j), for $j\geq 1,$ 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.

Received October 14 2003; revised version received January 19 2004. Published in Journal of Integer Sequences February 19 2004.

Partial Complements and Transposable Dispersions

Clark Kimberling and John E. Brown Department of Mathematics University of Evansville 1800 Lincoln Avenue Evansville, IN 47722 USA

Clark Kimberling and John E. Brown
Department of Mathematics
University of Evansville
1800 Lincoln Avenue
Evansville, IN 47722
USA