Permutations of the positive integers with restrictions on the sequence of differences

Peter John Slater and William Yslas Vélez

Vol. 71 (1977), No. 1, 193–196

DOI: 10.2140/pjm.1977.71.193

Abstract

Let {a_k} be a sequence of positive integers and d_k = |a_k+1 − a_k|. We say that {a_k} is a permutation if every positive integer appears once and only once in the sequence, {a_k}. We prove the following: Let {m_i} be any sequence of positive integers, then there exists a permutation {a_k} such that |{k|d_k = i}| = m_i.

Mathematical Subject Classification 2000

Primary: 05A05

Milestones

Received: 28 April 1976

Revised: 22 November 1976

Published: 1 July 1977