[Paper Review] Counting Irreducible Double Occurrence Words

J. C. Burns, Tilahun Muche|arXiv (Cornell University)|May 15, 2011

semigroups and automata theory9 references3 citations

TL;DR

This paper presents a unified combinatorial framework for counting irreducible, strongly irreducible, and palindromic double occurrence words over an n-letter alphabet, where each letter appears exactly twice. Using recursive decomposition and symmetry analysis, the authors derive closed-form formulas and new integer sequences, including a novel count for irreducible palindromes not previously listed in the OEIS.

ABSTRACT

A double occurrence word $w$ over a finite alphabet $Σ$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence words have been used for studying DNA recombination events. We develop formulas for counting and enumerating several elementary classes of double occurrence words such as palindromic, irreducible, and strongly-irreducible words.

Motivation & Objective

• To systematically classify and enumerate double occurrence words based on irreducibility and palindromic structure.
• To resolve the combinatorial challenge of counting non-isomorphic diagrams derived from double occurrence words using symmetry and reversal equivalence.
• To establish a new characterization of strongly irreducible double occurrence words through recursive decomposition.
• To identify and compute a previously unlisted integer sequence for irreducible palindromic words.
• To unify disparate counting results from topology, graph theory, and DNA recombination into a single coherent framework.

Proposed method

• Define double occurrence words as words of length 2n with each of n distinct letters appearing exactly twice.
• Use ascending order relabeling to normalize words and define equivalence classes, ensuring distinctness under relabeling.
• Classify words as palindromic if equivalent to their reverse; otherwise, classify as non-palindromic.
• Apply the formula |Total Diagrams| = (number of D.O. words + number of palindromes)/2 to count non-isomorphic diagrams.
• Employ recursive decomposition to build larger irreducible words from smaller ones, analyzing pointer sequences in macronuclear gene arrangements.
• Map micronuclear gene arrangements to double occurrence words via a homomorphism, identifying realizable words and detecting non-realizable cases.

Experimental results

Research questions

• RQ1 How can irreducible and strongly irreducible double occurrence words be systematically enumerated?
• RQ2 What is the precise count of palindromic double occurrence words, and how does it relate to the total number of words?
• RQ3 Which double occurrence words correspond to realizable micronuclear gene arrangements in ciliates?
• RQ4 Can a unified combinatorial framework be developed to count non-isomorphic diagrams derived from double occurrence words?
• RQ5 What new integer sequences emerge from counting irreducible palindromic double occurrence words, and are they novel?

Key findings

• The number of irreducible double occurrence words of length 2n is given by sequence A000698, with values such as 1, 2, 10, 74, 706, 8162 for n = 1 to 6.
• The number of strongly irreducible double occurrence words of length 2n is given by sequence A000699, with values such as 1, 1, 4, 27, 248, 2830 for n = 1 to 6.
• The count of irreducible palindromic double occurrence words forms a new integer sequence not previously listed in the OEIS, indicating a novel contribution.
• The total number of non-isomorphic diagrams derived from double occurrence words is given by (total D.O. words + palindromes)/2, enabling efficient enumeration.
• The shortest non-realizable double occurrence word is 11233244, confirming a boundary for realizable word generation from micronuclear gene arrangements.
• The homomorphism ϱ maps micronuclear gene arrangements to double occurrence words, with realizable words corresponding to valid pointer sequences in ciliate DNA recombination.