Generalized Wythoff arrays, shuffles and interspersions

✍ Scribed by A.S. Fraenkel; Clark Kimberling

• Book ID: 103060523
• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 683 KB
• Volume: 126
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90259-3

✦ Synopsis

Suppose a is a strictly increasing sequence of integers satisfying a(1) = 1. Let b be the ordered complement of a, and suppose b is infinite and satisfies b(n) 1. For every positive integer i, let w(i,j) be the sequence given by w(i, l)=a(a(i)), w(i,2)= b(a(i)), w(i,j)=a(w(i,j-1)) ifj is odd and 23, and w(i,j)= b(w(i,j-2)) ifj is even and 24. Then the sequences w(i, j), for i= 1,2,3, ,,. , partition the set of positive integers. For a(n) of the form [an], where a is an irrational number between 1 and 2, we denote the resulting array W(a). This is the original Wythoff array of a: =(l + fi)/2. For certain a, the row sequences of W(a) obey simple recurrence relations. For others, W(a) exhibits remarkable properties concerning the manner in which the terms of each row fit, in magnitude, among the terms of each other row. These properties are discussed in terms of shuffles and interspersions.