Goulden-Jackson Cluster Method Research Articles

A lifting of the Goulden–Jackson cluster method to the Malvenuto–Reutenauer algebra

A lifting of the Goulden–Jackson cluster method to the Malvenuto–Reutenauer algebra

From Hertzsprung’s problem to pattern-rewriting systems

Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation π∈𝒮 n contains the Hertzsprung pattern σ∈𝒮 k if there is a factor π(d+1)π(d+2)⋯π(d+k) of π such that π(d+1)-σ(1)=⋯=π(d+k)-σ(k). Using a combination of the Goulden–Jackson cluster method and the transfer-matrix method we determine the joint distribution of occurrences of any set of (incomparable) Hertzsprung patterns, thus substantially generalizing earlier results by Jackson et al. on the distribution of ascending and descending runs in permutations. We apply our results to the problem of counting permutations up to pattern-replacement equivalences, and using pattern-rewriting systems – a new formalism similar to the much studied string-rewriting systems – we solve a couple of open problems raised by Linton et al. in 2012.

Classical and consecutive pattern avoidance in rooted forests

Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern σ∈Sk if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as σ. For each permutation σ∈Sk−2, we construct a bijection between n-vertex forests avoiding (σ)(k−1)k≔σ(1)⋯σ(k−2)(k−1)k and n-vertex forests avoiding (σ)k(k−1)≔σ(1)⋯σ(k−2)k(k−1), giving a common generalization of results of West on permutations and Anders–Archer on forests. We further define a new object, the forest-Young diagram, which we use to extend the notion of shape-Wilf equivalence to forests. In particular, this allows us to generalize the above result to a bijection between forests avoiding {(σ1)k(k−1),(σ2)k(k−1),…,(σℓ)k(k−1)} and forests avoiding {(σ1)(k−1)k,(σ2)(k−1)k,…,(σℓ)(k−1)k} for σ1,…,σℓ∈Sk−2. Furthermore, we give recurrences enumerating the forests avoiding {123⋯k}, {213}, and other sets of patterns. Finally, we extend the Goulden–Jackson cluster method to study consecutive pattern avoidance in rooted trees as defined by Anders and Archer. Using the generalized cluster method, we prove that if two length-k patterns are strong-c-forest-Wilf equivalent, then up to complementation, the two patterns must start with the same number. We also prove the surprising result that the patterns 1324 and 1423 are strong-c-forest-Wilf equivalent, even though they are not c-Wilf equivalent with respect to permutations.

A generalized Goulden–Jackson cluster method and lattice path enumeration

The Goulden–Jackson cluster method is a powerful tool for obtaining generating functions counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths – both with height bounded and unbounded – by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued fraction formulas.

Generalizations of the Goulden–Jackson cluster method

We give several modifications of the Goulden–Jackson cluster method for finding generating functions for words avoiding a given set of forbidden words. Our modifications include functions which can take into account various ‘weights’ on words, including single letter probability distributions, double letter (i.e. pairwise) probability distributions, and triple letter probability distributions. We also describe an alternative, recursive approach to finding such generating functions. We describe Maple implementations of the various modifications. The accompanying Maple package is available at the website for this paper.

Generalized Correlation Functions and Their Applications in Selection of Optimal Multiple Spaced Seeds for Homology Search

The Goulden-Jackson cluster method is a powerful method to calculate the probability of occurrences of a pattern or set of patterns in a sequence. If the patterns contain wildcard characters, however, the size of the connector matrix grows exponentially with the number of wildcards. Here we show that average correlation c(z) is a good predicator of hitting probability q (n), and the generalized correlation function ĉ(z) can be used to approximate c(z) efficiently. We apply the method to the problem of optimal multiple spaced seed selection for homology search. We reexamine the concept of optimal sensitivity of spaced seeds and show that it is better to select optimal seeds based on some average properties, such as c(1), which is the expectation of the first hitting length. Higher order approximations can also be constructed easily. Tests on arbitrary large genomic data with multiple seeds show that the optimal multiple seeds selected by the methods are indeed more sensitive. The methods provide a theoretical background on which various empirical observations can be unified and further heuristic search methods can be developed.

Extension of Goulden–Jackson cluster method on pattern occurrences in random sequences and comparison with Régnier–Szpankowski method

The Goulden–Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2]. In this paper, we elaborate on one of the several extensions in [2], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2]. The extended result is then compared with the method of Régnier–Szpankowski [3], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier–Szpankowski method can be simplified to the extended Goulden–Jackson method.

Bounds for the growth rate of meander numbers

We provide improvements on the best currently known upper and lower bounds for the exponential growth rate of meanders. The method of proof for the upper bounds is to extend the Goulden–Jackson cluster method.

The symbolic Goulden-Jackson cluster method

Let V and B be a finite alphabet and a finite set of words (on V ), respectively. Suppose a(n) is the total number of words with length n that avoid the words in B as factors. The aim is to find the generating function in an efficient way. The Goulden-Jackson cluster method [Goulden, I. and Jackson, D.M., 1979, An inversion theorem for cluster decompositions of sequences with distinguished subsequences, Journal of the London Mathematical Society 20 (2), 567–576; Goulden, I. and Jackson, D.M., 1983, Combinatorial Enumeration (New York: John Wiley)], which is widely used in solving this kind of problems, has been beautifully explained, extended, and implemented by J. Noonan and D. Zeilberger (1999, The Goulden-Jackson cluster method: extensions, applications, and implementations, Journal of Difference Equations and Applications 5, 355–377). However, their Maple packages require that the cardinality of the alphabet is a numeric argument rather than symbolic. In this paper we extended the method into the latter case, thereby initiating the Symbolic Goulden-Jackson Method.

Exact Distribution of the Occurrence Number for K-tuples Over an Alphabet of Non-Equal Probability Letters

A nucleotide sequence can be considered as a realization of the non-equal-probability independently and identically distributed (niid) model. In this paper we derive the exact distribution of the occurrence number for each K-tuple with respect to the niid model by means of the Goulden-Jackson cluster method. An application of the probability function to get exact expectation curves [9] is presented, accompanied by comparison between the exact approach and the approximate solution.

Avoided Strings in Bacterial Complete Genomes and a Related Combinatorial Problem

The visualization of avoided and under-represented strings in some bacterial complete genomes raises a combinatorial problem which may be solved either by using the Goulden-Jackson cluster method or by construction of the minimal finite automaton defined by the set of forbidden words of the corresponding language.

The Goulden–Jackson Cluster Method for Cyclic Words

The powerful Goulden–Jackson Cluster Method, which generates generating functions enumerating words that avoid, as factors, a prescribed finite set of “mistakes,” is adapted to handle cyclic words (with marked beginnings).

Fractals from genomes – exact solutions of a biology-inspired problem

This is a review of a few recent papers with some new results added. After a brief biological introduction a visualization scheme of the string composition of long DNA sequences, in particular, of bacterial complete genomes, will be described. This scheme leads to a class of self-similar and self-overlapping fractals in the limit of infinitely long constituent strings. The calculation of their exact dimensions and the counting of true and redundant avoided strings at different string lengths turn out to be one and the same problem. We give exact solution of the problem using two independent methods: the Goulden–Jackson cluster method in combinatorics and the method of formal language theory.

The Goulden—Jackson cluster method: extensions, applications and implementations

Th powerful (and so far under-utilized) Goulden—Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of ‘dirty words’, is tutorialized and extended in various directions. The authors' Maple implementations, contained in several Maple packages available from this paper's website www.math.temple.edu/zeilberg/gj.html, ar described and explained.