Matjaz Konvalinka
In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially.
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The 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 (http://www.math.temple.edu/~zeilberg/gj.html), are described and explained.
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 th…
In this survey we discuss some of the significant contributions of Ian Goulden and David Jackson in the areas of classical enumeration, symmetric functions, factorizations of permutations, and algebraic foundations of quantum field theory. Through their groundbreaking textbook, {\em Combinatorial Enumeration}, and their numerous research papers, both together and with their many students, they have had an influence in areas of bioinformatics, mathematical chemistry, algorithmic computer science, and theoretical physics. Here we review and set in context highlights of their 40 years of collabor…
Let A be an alphabet and let F be a set of words with letters in A. We show that the sum of all words with letters in A with no consecutive subwords in F, as a formal power series in noncommuting variables, is the reciprocal of a series with all coefficients 0, 1 or -1. We also explain how this result is related to a result of Curtis Greene on lattices with Möbius function 0, 1, or -1.
The Goulden-Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper, we lift the cluster method for permutations to the Malvenuto-Reutenauer algebra. Upon applying standard homomorphisms, our result specializes to both the cluster method for permutations as well as a q-analogue which keeps track of the inversion number statistic. We construct additional homomorphisms using the theory of shuffle-compatibility, leading to furth…
We study the structure of the Goulden-Jackson-Vakil formula that relates Hurwitz numbers to some conjectural "intersection numbers" on a conjectural family of varieties $X_{g,n}$ of dimension $4g-3+n$. We give explicit formulas for the properly arranged generating function for these "intersection numbers", and prove that it satisfies Hirota equations. This generalizes and substantially simplifies our earlier results with Zvonkine in arXiv:math/0602457.
Shifted Macdonald Polynomials and the $(q,t)$-Deformed Goulden--Jackson Product
本文研究移位麦克唐纳多项式与(q,t)变形古尔德恩-杰克逊乘积，构造实现移位麦克唐纳本征值的无穷级数，推导纳扎罗夫-斯克莱宁算子相关级数并与已有算子比较。
The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for 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 f…
In 1996 Goulden and Jackson introduced a family of coefficients $( c_{π, σ}^λ ) $ indexed by triples of partitions which arise in the power sum expansion of some Cauchy sum for Jack symmetric functions $( J^{(α)}_π)$. The coefficients $ c_{π, σ}^λ $ can be viewed as an interpolation between the structure constants of the class algebra and the double coset algebra. Goulden and Jackson suggested that the coefficients $ c_{π, σ}^λ $ are polynomials in the variable $β:= α-1$ with non-negative integer coefficients and that there is a combinatorics of matching hidden behind them. This \emph{Matching…
We give two combinatorial proofs of Goulden and Jackson's formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type $B$.
Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Dołęga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^λ_{μ^0,...,μ^k}(b)$ obtained by the expansion of thes…
Constrained coding upper bounds via Goulden-Jackson cluster theorem
本文利用Goulden-Jackson聚类定理提出计算受限编码精确基数的新方法，揭示了其与谱方法的内在联系，并给出了变长非重叠码的显式上界。
We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula of Goulden, Jackson and Vakil for one part double Hurwitz numbers. Immediate consequences include a new proof that double Hurwitz numbers are piecewise polynomial, an understanding of the chamber structure and wall crossing for these polynomials, and a proof of the Goulden, Jackson and Vakil's Strong Piecewise Polynomiality conjecture. The method is a straightforward application of Okounkov's expression for double Hurwitz numbers in terms of operat…
We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists, and establish the semi-rigorous bound |freq_1(K) - 1/2| <= 1/46.
Goulden and Jackson first introduced the cut-and-join operator. The cut-and-join is widely used in studying the Hurwitz number and many other topological recursion problems. Mironov, Morosov and Natanzon give a more general construction and call it W-operator W([n]). As a special case, the cut-and-join operator is W([2]). In this paper, we study the structure of W([n]). We prove that W([n]) can be written as the sum of n! terms and each term corresponds uniquely to a permutation in Sn. We also define the degree of each term. We prove that there is a correspondence between the terms of W([n]) w…
In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if it is ever to be constructed) by methods close to those of M. Kazarian and S. Lando in [7]. In particular, we prove a Witten-Kontsevich-type theorem relating the intersection theory and integrable hierarchies. We also extend the results of [7] to include the Hodge integrals over the moduli spaces, in…
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