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arXiv · 1710.05651

The Nonassociativity of the Double Minus Operation

Jia Huang·Madison Mickey·Jianbai Xu
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Abstract

The sequence A000975 in OEIS can be defined by $A_1=1$, $A_{n+1}=2A_n$ if $n$ is odd, and $A_{n+1}=2A_n+1$ 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 $a\ominus b := -a -b$. We show that the number of distinct results obtained by inserting parentheses in the expression $x_0\ominus x_1\ominus \cdots\ominus x_n$ equals $A_n$, by investigating the leaf depth in binary trees. Our result can be viewed as a quantitative measurement for the nonassociativity of the binary operation $\ominus$.

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Jia Huang, Madison Mickey, Jianbai Xu. 2017-10-16. The Nonassociativity of the Double Minus Operation. https://arxiv.org/abs/1710.05651

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