Enumerating k-connected blocks and gk-connected blocks in words

We define two new statistics on words: the k-connector and the gk-connector. For a word $\pi ={\pi }_1{\pi }_2\cdots {\pi }_n$ of length $n$ over the alphabet $[k]$, a k-connector is defined as an ordered pair $({\pi }_j,{\pi }_{j+1})$ where $1\le j\le n-1$ and ${\pi }_j+{\pi }_{j+1}=k$. Conversely, a gk-connector is defined as an ordered pair $({\pi }_j,{\pi }_{j+1})$ where $1\le j\le n-1$ and ${\pi }_j+{\pi }_{j+1}>k$. We investigate the enumeration of partitions based on these statistics, providing generating functions and explicit formulas.