Up-Down Sequences: Inversions, Coinversions, and the Sum of Major Indices

Abstract

For $n\ge 1$ we call a sequence $s_1,s_2,\dots ,s_n$ an up-down sequence of length $n$ when (i) $s_1=1$; (ii) $s_i\in \{1,2,3,4\}$, for $2\le i\le n$; and, (iii) $|s_i–s_{i-1}|=1$, for $2\le i\le n$. We count the number of inversions and coinversions for all such up-down sequences of length $n$, as well as the sum of the major indices for all these sequences of length $n$.