# Counting pairs of words according to the number of common rises, levels, and descents

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review

## Abstract

A level (L) is an occurrence of two consecutive equal entries in a word w = w1w2 . . . , while a rise (R) or descent (D) occurs when the right or left entry, respectively, is strictly larger. If u = u1u2 . . . un and v = v1v2 . . . vn are k-ary words of the same given length and 1 ≤ i ≤ n-1, then there is, for example, an occurrence of LR at index i if ui = ui+1 and vi < vi+1, and, likewise, for the other possibilities. Similar terminology may be used when discussing ordered d-tuples of k-ary words of length n (the set of which we'll often denote by [k]nd). In this paper, we consider the problem of enumerating the members of [k]nd according to the number of occurrences of the pattern ρ, where d ≥ 1 and ρ is any word of length d in the alphabet {L,R,D}. In particular, we find an explicit formula for the generating function counting the members of [k]nd according to the number of occurrences of the patterns ρ = LiRd-i, 0 < i < d, which, by symmetry, is seen to solve the aforementioned problem in its entirety. We also provide simple formulas for the average number of occurrences of ρ within all of the members of [k]nd, providing both algebraic and combinatorial proofs. Finally, in the case d = 2, we solve the problem above where we also allow for weak rises (which we'll denote by Rw), i.e., indices i such that wi ≤ wi+1 in w. Enumerating the cases RwRw and RRw seems to be more difficult, and to do so, we combine the kernel method with the simultaneous use of several recurrences.

Original language English 62 Online Journal of Analytic Combinatorics 9 Published - 2014

## Keywords

• Descent
• Generating function
• K-ary word
• Level
• Rise

## ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics

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