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This is a Quantitative Study study.

October 1, 2025

Counting adjacencies with difference at most one in -ary words

Key Points

The study computes a generating function for the number of adjacencies in -ary words, revealing combinatorial properties.
Special cases involve staircase and Hertzsprung words, with results expressed in terms of Chebyshev polynomials.
Analyses extend to a circular case where the boundary condition is considered, broadening the implications of the findings.
The results apply to finite set partitions, providing a deeper link between these combinatorial concepts.

Abstract

In this paper, we study the distribution on \ (kⁿ\) for the parameter recording the number of indices \ (i n-1\) within a word \ (w=w₁ wₙ\) such that \ (|w₈+₁-wᵢ|\ 1\) and compute the corresponding (bivariate) generating function. A circular version of the problem wherein one considers whether or not \ (|wₙ-w₁|\ 1\) as well is also treated. As special cases of our results, one obtains formulas involving staircase and Hertzsprung words in both the linear and circular cases. We make use of properties of special matrices in deriving our results, which may be expressed in terms of Chebyshev polynomials. A generating function formula is also found for the comparable statistic on finite set partitions with a fixed number of blocks represented sequentially.

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