[Paper Review] Counting the occurrences of generalized patterns in words generated by a morphism

Sergey Kitaev, Toufik Mansour|ArXiv.org|Oct 11, 2002

semigroups and automata theory19 references3 citations

TL;DR

This paper develops a method to count occurrences of generalized permutation patterns—such as 1-2, 2-1, 123, and 1-1-...-1—in finite approximations of morphic sequences. Using a morphism-based iterative construction (e.g., 1→123, 2→13, 3→2), the authors derive exact formulas for pattern counts, showing, for example, that the number of 1-2 patterns in the n-th iterate is $3 \cdot 4^{n-1} + 2^n$. The key contribution is a systematic framework for counting pattern occurrences in morphic words via structural decomposition and recurrence relations.

ABSTRACT

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a morphism generating a nonrepetitive sequence.

Motivation & Objective

• To develop a method for counting occurrences of generalized permutation patterns in finite approximations of morphic sequences.
• To extend classical pattern counting to include generalized patterns without internal dashes and with repeated letters.
• To provide exact closed-form formulas for pattern counts in specific morphic sequences, such as the nonrepetitive sequence generated by 1→123, 2→13, 3→2.
• To establish conditions under which external pattern occurrences can be excluded, enabling recursive counting via structural decomposition.

Proposed method

• The authors define morphisms on finite alphabets and generate words via iteration of a fixed point, focusing on finite approximations $\phi^n(1)$.
• They introduce the concept of external occurrences of a pattern $\tau$ across subwords $X_i$ and $X_j$, which are substrings in the image of the morphism.
• For patterns without internal dashes, they derive a recurrence: $N^\tau_\phi(n) = (d + \ell)^{n-2} \sum_{i=1}^k s_i$ for $n \geq 2$, where $s_i$ counts internal occurrences in $X_i$, and $d, \ell$ are parameters from the morphism’s structure.
• They use combinatorial arguments based on the number of times each subword appears in the iterated image, leveraging binomial coefficients for selection of pattern positions.
• The method relies on verifying the absence of external pattern occurrences across pairs of subwords, which ensures the recurrence holds.
• They validate the method with examples, including the morphism $\phi_w: 1\to123, 2\to13, 3\to2$, and derive explicit formulas for patterns like 12, 123, and 21.

Experimental results

Research questions

• RQ1 How can one count the number of occurrences of a generalized pattern in finite approximations of a morphic sequence?
• RQ2 What conditions ensure that pattern occurrences in morphic words can be counted via a recurrence relation?
• RQ3 For the classical morphism generating a nonrepetitive sequence (1→123, 2→13, 3→2), what is the exact number of 1-2 pattern occurrences in the n-th iterate?
• RQ4 Can the method be applied to patterns like 123 or 21, even when external occurrences complicate the count?
• RQ5 What is the role of subword structure and morphism parameters (e.g., $d$, $\ell$) in determining the growth rate of pattern occurrences?

Key findings

• The number of 1-2 pattern occurrences in the n-th iterate of the morphism $\phi_w$ is $3 \cdot 4^{n-1} + 2^n$.
• For the pattern 12 (i.e., rises), the number of occurrences in $\phi_w^n(1)$ is $3 \cdot 2^{n-2}$ for $n \geq 2$, with 2 occurrences at $n=1$.
• The number of 123 pattern occurrences in $\phi_w^n(1)$ is $2^{n-2}$ for $n \geq 2$, and zero for $n=1$.
• The number of 21 pattern occurrences (descents) in $\phi_w^n(1)$ is $3 \cdot 2^{n-2} - 1$, derived from total length minus rises minus one.
• For the pattern $1\mbox{-}1\mbox{-}1\mbox{-}1$, the number of occurrences in $\phi_w^n(1)$ is $3 \cdot \binom{2^{n-1}}{4}$ for $n \geq 3$, and zero for $n=1,2$.
• The method applies to generalized patterns without internal dashes when external occurrences are absent, and yields a recurrence based on the sum of internal pattern counts across morphism images.