Eulerian pairs on Fibonacci words

Details Eulerian pairs on Fibonacci words Eulerian pairs on Fibonacci words

2011-10-02

arxiv.org/abs/1110.0189v1

Mathematics

Combinatorics

14 pages

Scientific paper

In their recent study of Mahonian pairs, Sagan and Savage introduced the notion of Eulerian pairs. A pair $(S,T)$ of two finite sets of words is said to be an Eulerian pair if the distribution of the descent number over $S$ equals the distribution of the excedance number over $T$. Let $\Phi_1$ denote Foata's first fundamental transformation and $\Psi$ denote a bijection of Han on words. We observe that $\Phi_1$ and $\Psi$ coincide when restricted to words on $\{1,2\}$. Using the inverse of $\Phi_1$ or $\Psi$ for words on $\{1, 2\}$, we obtain Eulerian pairs on Fibonacci words, where a Fibonacci word is defined to be a word on $\{1,2\}$ with no consecutive ones. By modifying a bijection of Steingr\'{\i}msson, we find another Eulerian pair on Fibonacci words.

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