Enumeration of consecutive patterns in flattened Catalan words
Mark Shattuck
TL;DR
This work studies the distribution of consecutive subword patterns in flattened Catalan words, a refined subset of Catalan words with a nondecreasing sequence of initial letters from each increasing run. It develops a multivariate generating-function framework that tracks length, an auxiliary run-based statistic (trun) and pattern occurrences, yielding explicit closed forms for joint distributions of various pattern triples and their totals or avoidances. Notable contributions include exact generating functions that capture ascents/descents/levels and several triples of length two or three, as well as identified distributional equivalences such as 112 ≈ 122 and 211 ≈ 221 ≈ 231, supported by combinatorial proofs. The paper further extends the methodology to additional pattern triples (3.1–3.3), delivering explicit formulas and highlighting the broad applicability of the approach to pattern statistics on flattened Catalan words.
Abstract
A Catalan word $w$ is said to be flattened if the subsequence of $w$ obtained by taking the first letter of each weakly increasing run is nondecreasing. Let $\mathcal{F}_n$ denote the set of flattened Catalan words of length $n$, which has cardinality $\frac{3^{n-1}+1}{2}$ for all $n \geq 1$. In this paper, we consider the distribution of several consecutive patterns on $\mathcal{F}_n$. Indeed, we find explicit formulas for the generating functions of the joint distribution on $\mathcal{F}_n$ of several trios of patterns, along with an auxiliary parameter. As special cases of these formulas, we obtain the generating function for the distribution of all consecutive patterns of length two or three. The following equivalences with regard to being identically distributed on $\mathcal{F}_n$ arise when comparing the various generating functions and may be explained bijectively: $112\approx122$ and $211\approx221\approx231$. In addition, explicit expressions are found for the total number of occurrences on $\mathcal{F}_n$ of each pattern of length two or three as well as for the number of avoiders of each pattern. These results can be obtained as special cases of our more general formulas for the generating functions, but may be explained combinatorially as well, the arguments of which are featured herein.