Further results on staircase (cyclic) words

TL;DR

The paper addresses counting staircase words over a $k$-letter alphabet and their cyclic variants by deriving two-variable generating functions $F_k(x,t)$ and $G_k(x,t)$ for the distributions of adjacencies bounded by 1, expressing them in closed form via Chebyshev polynomials. The authors employ matrix formulations, the Sherman–Morrison formula, and Chebyshev identities to obtain $F_k(x,t)=1/(1-x\gamma(x,t))$ with a defined $\gamma(x,t)$ and parameter $\phi$, and to handle per-prefix counts through $f_i(x,t)$ and $A(x,t)f(x,t)=x\mathbf{1}$. The cyclic case is treated with a similar matrix-analytic framework to yield $G(x,t)$ and $g_{i,j}(x,t)$, along with aggregated generating functions $E(x)$ and cyclic Hertzsprung variants; the work also provides explicit small-$k$ formulas. Overall, the results extend prior one-variable analyses, offering explicit, algebraic generating functions and new connections to Hertzsprung-type word classes that facilitate precise enumerations and asymptotics.

Abstract

We find the two-variables generating function for the statistic which counts the number of variations in a word bounded by $1$. Thus, we refine and extend previous results concerning staircase words, which are words in which the variation between all consecutive letters is bounded by $1$. We obtain the analogue results for cyclic words.

Theorems & Definitions (14)

• Theorem 1
• proof
• Corollary 2
• proof
• Definition 3
• Corollary 4
• proof
• Theorem 5
• proof
• Example 6