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 $1 \leq j \leq n-1$ and $\pi_j + \pi_{j+1} = k$. Conversely, a gk-connector is defined as an ordered pair $(\pi_j, \pi_{j+1})$ where $1 \leq j \leq n-1$ and $\pi_j + \pi_{j+1} > k$. We investigate the enumeration of partitions based on these statistics, providing generating functions and explicit formulas.