Prefixes of Stanley’s Catalan Paths with Odd Returns to the \(x\)-axis–Standard Version and Skew Catalan-Stanley Paths

For \(r=1,2,…, 6\), we obtain generating functions \(F^{(r)}_{k}(y)\) for words over the alphabet \([k]\), where \(y\) tracks the number of parts and \([y^n]\) is the total number of distinct adjacent \(r\)-tuples in words with \(n\) parts. In order to develop these generating functions for \(1\le r\le 3\), we make use of intuitive decompositions but for larger values of \(r\), we switch to the cluster analysis method for decorated texts that was introduced by Bassino et al. Finally, we account for the coefficients of these generating functions in terms of Stirling set numbers. This is done by putting forward the full triangle of coefficients for all the sub-cases where \(r=5\) and 6. This latter is shown to depend on both periodicity and number of letters used in the \(r\)-tuples.