Counting up-up-or-down-down permutations

TL;DR

This paper derives a bivariate exponential generating function for a specific class of permutations called up-up-or-down-down permutations, answering a question posed by Donald Knuth about their enumeration based on last entry.

Contribution

It introduces the first explicit generating function for up-up-or-down-down permutations of odd length, characterizing their structure and enumeration.

Findings

Derived the bivariate exponential generating function for the permutations

Characterized permutations where peaks and valleys are odd

Provided enumeration results for these permutations

Abstract

Answering a question of Donald Knuth, we find the bivariate exponential generating function for "up-up-or-down-down'' permutations of odd length according to their last entry. An up-up-or-down-down permutation is a permutation $a_1a_2\cdots a_n$ satisfying $a_{2i-1}<a_{2i}$ if and only if $a_{2i}<a_{2i+1}$ for $1\le i <n/2$. Equivalently, an up-up-or-down-down permutation is one in which every peak and every valley is odd.