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

For \( n \geq 1 \) we call a sequence \( s_1, s_2, \ldots, s_n \) an up-down sequence of length \( n \) when (i) \( s_1 = 1 \); (ii) \( s_i \in \{1, 2, 3, 4\} \), for \( 2 \leq i \leq n \); and, (iii) \( |s_i – s_{i-1}| = 1 \), for \( 2 \leq i \leq n \). 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 \( n \).