# Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons

## TL;DR

This paper explores the enumeration of cyclic orders with prescribed triple relations, introduces multidimensional boustrophedon methods, and offers new insights into Euler and Entringer numbers.

## Contribution

It develops multidimensional boustrophedon techniques to enumerate cyclic orders and provides novel interpretations of classical combinatorial numbers.

## Key findings

- Enumeration formulas for cyclic orders with prescribed triples
- Introduction of multidimensional boustrophedon methods
- New interpretations for Euler and Entringer numbers

## Abstract

We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the enumeration of descent classes of permutations, which is done via the boustrophedon construction. In other cases, we solve the question by introducing multidimensional versions of the boustrophedon. In particular we find new interpretations for the Euler up/down numbers and the Entringer numbers.