The nonassociativity of the double minus operation

Jia Huang, Madison Mickey, Jianbai Xu

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Abstract

The sequence A000975 in the Encyclopedia of Integer Sequences can be defined by A1 = 1, An+1 = 2An if n is odd, and An+1 = 2An+1 if n is even. This sequence satisfies other recurrence relations, admits some closed formulas, and is known to enumerate several interesting families of objects. We provide a new interpretation of this sequence using a binary operation defined by a ⊖ b := -a - b. We show that the number of distinct results obtained by inserting parentheses in the expression x0 ⊖ x1 ⊖ … ⊖ xn equals An, by investigating the leaf depth in binary trees. Our result can be viewed as a quantitative measurement for the nonassociativity of the binary operation ⊖.

Original languageEnglish (US)
Article number17.10.3
JournalJournal of Integer Sequences
Volume20
Issue number10
Publication statusPublished - Jan 1 2017

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Keywords

  • Binary sequence
  • Binary tree
  • Leaf depth
  • Nonassociativity
  • Parenthesization

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics

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