Young Graphs: 1089 et al.
L. H. Kendrick
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139
This paper deals with those positive integers N such that, for
given integers g and k with 2 ≤ k < g, the
base-g digits of kN appear in reverse order from those of
N. Such N are called (g, k) reverse
multiples. Young, in 1992, developed a kind of tree reflecting
properties of these numbers; Sloane, in 2013, modified these trees into
directed graphs and introduced certain combinatorial methods to
determine from these graphs the number of reverse multiples for given
values of g and k with a given number of digits. We prove
Sloanes isomorphism conjectures for 1089 graphs and complete graphs,
namely that the Young graph for g and k is a 1089 graph
if and only if k+1 | g and is a complete Young graph on
m nodes if and only if ⌊ gcd(g - k, k2 - 1)/(k + 1) ⌋ = m - 1. We also extend his study of cyclic
Young graphs and prove a minor result on isomorphism and the nodes
adjacent to the node [0, 0].
Full version: pdf,
(Concerned with sequences
Received April 10 2015;
revised version received August 16 2015; August 20 2015.
Published in Journal of Integer Sequences, August 21 2015.
Journal of Integer Sequences home page