Enumerating k-connected blocks and gk-connected blocks in words
Walaa Asakly, Noor Kezil
TL;DR
The paper defines two new word statistics, $k$-connectors (pairs with sum $k$) and $g_k$-connectors (pairs with sum $>k$, both on words over $[k]$), and develops generating-function techniques to enumerate words by these statistics. It derives a closed-form generating function for $k$-connectors, $C_k(x,q)=\frac{1}{1-x-\left( \frac{x+x^2(q-1)}{1-x^2(q-1)^2}\right)(k-1)}$, and proves the total number of $k$-connector blocks among words of length $n$ is $(k-1)(n-1)k^{n-2}$. For $g_k$-connectors, the paper constructs a determinant-based framework to obtain $GC_k(x,q)$, with explicit even/odd $k$ formulas, yielding that the number of $g_k$-connector blocks is $(k+1)/2\,(n-1)k^{n-1}$ for even $k$ and $\lfloor\frac{k+1}{2}\rfloor\,k^{n-1}(n-1)$ for odd $k$, and provides derivative-based counts via $\frac{\partial GC_k(x,q)}{\partial q}\big|_{q=1}$ revealing a $\frac{k(k+1)}{2}\frac{x^2}{(1-kx)^2}$ contribution. These results advance the combinatorics of word statistics, offering explicit generating-function and determinant-based methods for analyzing adjacent-sum patterns in words.
Abstract
We define two new statistics on words: the k-connector and the gk-connector. For a word $π= π_1π_2\cdotsπ_n$ of length $n$ over the alphabet $[k]$, a k-connector is defined as an ordered pair $(π_j, π_{j+1})$ where $1 \leq j \leq n-1$ and $π_j + π_{j+1} = k$. Conversely, a gk-connector is defined as an ordered pair $(π_j, π_{j+1})$ where $1 \leq j \leq n-1$ and $π_j + π_{j+1} > k$. We investigate the enumeration of partitions based on these statistics, providing generating functions and explicit formulas.