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

Walaa Asakly; Noor Kezil

paper

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

arXiv:2503.14073

Abstract

We define two new statistics on words: the k-connector and the gk-connector. For a word $I ¨ = I ¨ _{1} I ¨ _{2} \dots I ¨ _{n}$ of length $n$ over the alphabet $[k]$ , a k-connector is defined as an ordered pair $(I ¨ _{j}, I ¨ _{j + 1})$ where $1 \leq j \leq n - 1$ and $I ¨ _{j} + I ¨ _{j + 1} = k$ . Conversely, a gk-connector is defined as an ordered pair $(I ¨ _{j}, I ¨ _{j + 1})$ where $1 \leq j \leq n - 1$ and $I ¨ _{j} + I ¨ _{j + 1} > k$ . We investigate the enumeration of partitions based on these statistics, providing generating functions and explicit formulas.