Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons

Sanjay Ramassamy

Published 2017-06-11Version 1

We enumerate total cyclic orders on $\left\{x_1,\ldots,x_n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(x_i,x_{i+1},x_{i+2})$, with indices 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.