The aim of this project is to systematize (asymptotic) coefficient extraction from a wide class of naturally occurring multivariate generating functions. We aim to take a genuinely multivariate approach, and to improve over previous work in the areas of generality, ease of use, and suitability for effective computation. The topic is inherently interdisciplinary and uses complex analysis in one and several variables, asymptotics of integrals, topology, algebraic geometry, and symbolic computation. The application areas are many: we are particularly motivated by specific naturally occurring problems arising from areas such as multivariate recurrence relations, random tilings, queueing theory, and analysis of algorithms and data structures. If you are interested in working on these problems, please let us know.

Our methods use complex analysis and oscillatory integrals to analyse singularities of explicitly known generating functions. We deal with generating functions of the form $F(\mathbf{z}) = \sum a_{r_1}\dots a_{r_d} z_1^{r_1} \cdots z_d^{r_d}$ which can locally be expressed in the form $G/H$ with $G$ and $H$ analytic functions. The singular set of this function (zero-set of $H$) plays a large role.

The long-term organizers of this project are: Robin Pemantle | Mark C. Wilson | Yuliy Baryshnikov | Stephen Melczer | Marni Mishna .

## Events related to this project

• An American Mathematical Society Mathematics Research Community, with a main workshop in Summer 2021 (delayed by covid from 2020) and follow-up activities for the following year, including an invited session at the 2022 Joint Mathematics Meetings.
• A workshop at the American Institute of Mathematics (AIM). Orginally scheduled for Summer 2021 and delayed by covid until Summer 2022.
• ## Publications in this project

Publications are generally listed in reverse chronological order of production. Papers applying results of this project to other areas are listed below this section. Versions linked here are usually not the official published versions, but they are usually the latest prepublication versions held by the authors.

### Main References

• Analytic Combinatorics in Several Variables by Robin Pemantle and Mark C. Wilson. Cambridge University Press, 2013. (Second edition expected late 2021 or 2022)

Textbook containing the most complete and expansive treatment of the theory for researchers.

• An Invitation to Analytic Combinatorics: From One to Several Variables by Stephen Melczer. Texts & Monographs in Symbolic Computation, Springer International Publishing, 2021.

Account of the theory aimed at graduate students and those looking to enter the field. More computational focus.

• ### Publications concerning core theory of the project

1. Analytic Combinatorics in Several Variables by Robin Pemantle, Mark C. Wilson, Stephen Melczer. Forthcoming 2nd edition of Cambridge monograph, Expected late 2021.
2. An Invitation to Analytic Combinatorics: From One to Several Variables by Stephen Melczer. Texts & Monographs in Symbolic Computation, Springer International Publishing, 2021.
3. Stationary points at infinity for analytic combinatorics by Yuliy Baryshnikov, Stephen Melczer and Robin Pemantle. Under review 2021.

On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable conditions under which the behavior at infinity may be ignored, and the usual theorems of classical and stratified Morse theory may be applied. Maple code for this paper

4. Asymptotics of multivariate sequences in the presence of a lacuna by Yuliy Baryshnikov, Stephen Melczer and Robin Pemantle. Under review 2021.

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value, in even dimensions at least 4. This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and GÃ¤rding. As a consequence, we give a topological explanation for certain asymptotic phenomenon appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions. This in turn enables the rigorous determination of integer coefficients in the Morse-Smale complex, which are difficult to determine using direct geometric methods.

5. Effective Coefficient Asymptotics of Multivariate Rational Functions via Semi-Numerical Algorithms for Polynomial Systems by Bruno Salvy and Stephen Melczer. Journal of Symbolic Computation, Volume 103, 234–279, 2021.

This subsumes the results in the earlier conference paper (ISSAC 2016). The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

6. Analytic Combinatorics: A Multidimensional Approach by Marni Mishna. Discrete Mathematics and its Applications, CRC Press.
7. Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration by Stephen Melczer. PhD thesis (Waterloo/Lyon), 259 pages.

We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.

8. Asymptotics of Bivariate Generating Functions with Algebraic Singularities by Torin Greenwood. J. Combinatorial Theory A 2018.

Deals with bivariate algebraic singularities.

9. Analytic Combinatorics in Several Variables by Robin Pemantle and Mark C. Wilson. Cambridge University Press, 2013.

Textbook containing the most complete and expansive treatment of the theory.

10. Automatic asymptotics for coefficients of smooth, bivariate rational functions by Timothy DeVries, Joris van der Hoeven and Robin Pemantle. Online J. Anal. Comb., vol. 6, 24 pages (2012).

Shows how to compute contributing points algorithmically even when they are not minimal, extending the applicability of the smooth point methods.

11. New software for computing asymptotics of multivariate generating functions by Alexander Raichev. ACM Communications in Computer Algebra 45 (2011), 183-185.

Describes the Sage package amgf .

12. Asymptotics of coefficients of multivariate generating functions: improvements for smooth points by Alexander Raichev and Mark C. Wilson. Online Journal of Analytic Combinatorics, 2011.

We improve over the first paper by Pemantle and Wilson, and derive explicit formulae for the entire asymptotic series obtained by analysis near a smooth point of the singular variety. The numerical results obtained show how well these approximations work in practice.

13. Analytic combinatorics in $d$ variables: An overview by Robin Pemantle. AMS Contemporary Mathematics 520.

14. A case study in bivariate singularity analysis by Timothy Devries. AMS Contemporary Mathematics 520.

The first detailed treatment of an example in which asymptotics in the scale of $n^{1/4}$ occur rather than the usual $n^{1/2}$. A degenerate saddle point occurs in a natural counting problem whose answer is expressed as the diagonal sequence of a bivariate rational generating function. Results from previous papers do not cover this case. Performing the analysis in this example represents a first step towards understanding general cases of this geometric type.

15. Asymptotic expansions of oscillatory integrals with complex phase by Robin Pemantle and Mark C. Wilson. AMS Contemporary Mathematics 520.

We consider saddle point integrals in $d$ variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The method is via contour shifting in complex $d$-space. This work is motivated by applications to asymptotic enumeration, and fulfils a promised citation made in an earlier paper by these authors.

16. Asymptotics of multivariate sequences, part III: Quadratic points by Yuliy Baryshnikov and Robin Pemantle. Advances in Mathematics 228 (2011), 3127-3206.

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist cone points, near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems. This deals with another case in the original taxonomy outlined in the first paper of Pemantle and Wilson.

Note: original title was "Tilings, groves and multiset permutations: asymptotics of rational generating functions whose pole set includes a cone".
17. Asymptotics of coefficients of multivariate generating functions: improvements for smooth points by Alexander Raichev and Mark C. Wilson. Electron. J. Combin. 15 (2008), no. 1, Research Paper 89, 17 pp.

We improve over the first paper by Pemantle and Wilson, and derive explicit formulae for the entire asymptotic series obtained by analysis near a smooth point of the singular variety. The numerical results obtained show how well these approximations work in practice.

18. A new method for computing asymptotics of diagonal coefficients of multivariate generating functions by Alexander Raichev and Mark C. Wilson. Proceedings of International Conference on Analysis of Algorithms, Juan-les-Pins, 2007.

Let $\sum_{\mathbf{n}\in\mathbb{N}^d} f_\mathbf{n} \mathbf{x^n}$ be a multivariate generating function that converges in a neighborhood of the origin of $\mathbb{C}^d$. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients $f_{a_1n,\ldots,a_dn}$ and show its superiority over the standard, univariate diagonal method.

19. Mixed powers of generating functions by Manuel Lladser. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, Discrete Mathematics and Theoretical Computer Science Proceedings, AG, 171-182, 2006.

This paper derives uniform asymptotics for a multivariate generalization of Riordan arrays, namely a GF of the form $\prod_j (1 - w_j f_j(z))^{-1}$. In other words we seek the asymptotics of $[z^{n_0}] \prod_j f_j(z)^{n_j}$ as at least one of the exponents $n_j$ tends to infinity. This generalizes several results in the literature. Once again reduction to Fourier-Laplace integral is the key step.

20. Uniform formulae for coefficients of meromorphic functions in two variables. Part I by Manuel Lladser. SIAM Journal on Discrete Mathematics 20 (2007), 811-828.

This paper derives uniform asymptotics in the smooth bivariate case when the directions in question are such that there is no phase change in the underlying Fourier-Laplace integral. This generalizes results in the first Pemantle-Wilson paper.

21. Twenty combinatorial examples of asymptotics derived from multivariate generating functions by Robin Pemantle and Mark C. Wilson. SIAM Review 50 (2008), 199-272.

We illustrate the use of the techniques of some of the papers below, on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morse-theoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have the property that generating functions derived from them are amenable to our asymptotic analyses, and we describe further machinery that facilitates computations for these classes of examples.

22. Asymptotics for generalized Riordan arrays by Mark C. Wilson. Discrete Mathematics and Theoretical Computer Science, volume AD (2005), 323-334 (Proceedings of the 2005 International Conference on Analysis of Algorithms, Barcelona).

This applies the results of papers below to derive bivariate asymptotics for the special case $F(x, y) = \phi(x)/(1 - yv(x))$, which arises frequently in applications.

23. Convolutions of inverse linear functions via multivariate residues by Yuliy Baryshnikov and Robin Pemantle. Preprint (2004), 42 pages.

This uses stratified Morse theory to derive complete expansions when $V$ is a union of hyperplanes ($H$ is a product of linear factors). Such GFs occur often in queueing theory. It does not handle directions on the boundary of the cones mentioned below (Asymptotics of Multivariate Sequences, II). At present, each of these articles does something the other cannot.

24. Asymptotics of multivariate sequences II: multiple points of the singular variety by Robin Pemantle and Mark C. Wilson. Combinatorics, Probability and Computing 13 (2004), 735-761.

In this article we deal with multiple points of $V$. Our results show that the central limit (Ornstein-Zernike) behavior typical of the smooth case does not hold in the multiple point case. For example, when $\mathcal{V}$ has a multiple point singularity at $(1 , \ldots , 1)$, rather than $a_\mathbf{r}$ decaying as $|\mathbf{r}|^{-1/2}$ as $|\mathbf{r}| \to \infty$, $a_\mathbf{r}$ is very nearly polynomial in a cone of directions.

25. Asymptotic enumeration via singularity analysis by Manuel Lladser. PhD thesis, Ohio State University 2003.

Asymptotics for smooth points are computed in the above articles by Fourier-Laplace integrals, and are usually uniform for most sets of directions. However there can be directions where the asymptotic behaviour changes (for example, in "Airy phenomena" where $r^{-1/2}$ becomes $r^{-1/3}$) and uniformity breaks down. The thesis studies such situations in detail in the two-dimensional case. It relies on the asymptotic analysis of a certain type of "parameter-varying" F-L integral of the form $\int \exp(-s P(d,x)) A(d,x) \, dx$. The cases of interest are when either the phase term $P(d,x)$ or the amplitude term $A(d,x)$ exhibits a change of degree as $d$ approaches a degenerate direction. These are handled by a generalized version of the stationary phase and the coalescing saddle point method. Uniform expansions and local limit results are obtained.

26. Asymptotics of multivariate sequences I: smooth points of the singular variety by Robin Pemantle and Mark C. Wilson, Journal of Combinatorial Theory A 97 (2002), 129-161.

This article treats the case of smooth points of $V$. It includes a detailed introduction to the project, discusses previous work on the topic and lays the groundwork for further papers by devising a taxonomy. Then it treats the most commonly occurring case, smooth points. Companion papers will treat points of $V$ where the local geometry is more complicated, and for which other methods of analysis are not known.

Note: the published version (reprints are available) contains many errors, several the fault of the publisher. In particular, in Thm 3.5 there should be a minus sign in the denominator of the expression for the constant C_0.

27. Generating functions with high-order poles are nearly polynomial by Robin Pemantle. Mathematics and computer science (Versailles, 2000), 305--321. Trends in Mathematics, Birkhauser, Basel, 2000.

Explains why our multivariate asymptotic expansions sometimes have only finitely many terms, and hence are effectively computable.

### Papers using the basic theory in application areas

1. Asymptotic Enumeration of Lonesum Matrices Jessica Khera, Erik Lundberg, and Stephen Melczer. Advances in Applied Mathematics, Volume 123, 102–118, 2021.

We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval's identity. In addition, we provide an application in Algebraic Statistics on the asymptotic ML-degree of the bivariate multinomial missing data problem, and strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.

2. Counting walks with large steps in an orthant Alin Bostan, Mireille Bousquet-Mélou, and Stephen Melczer. Accepted to Journal of the European Mathematical Society, 2020.

In the past fifteen years, the enumeration of lattice walks with steps taken in a prescribed set S and confined to a given cone, especially the first quadrant of the plane, has been intensely studied. As a result, the generating functions of quadrant walks are now well-understood, provided the allowed steps are small, that is S in {−1,0,1}^2. In particular, having small steps is crucial for the definition of a certain group of bi-rational transformations of the plane. It has been proved that this group is finite if and only if the corresponding generating function is D-finite (that is, it satisfies a linear differential equation with polynomial coefficients). This group is also the key to the uniform solution of 19 of the 23 small step models possessing a finite group. In contrast, almost nothing is known for walks with arbitrary steps. In this paper, we extend the definition of the group, or rather of the associated orbit, to this general case, and generalize the above uniform solution of small step models. When this approach works, it invariably yields a D-finite generating function. We apply it to many quadrant problems, including some infinite families. After developing the general theory, we consider the 13 110 two-dimensional models with steps in {−2,−1,0,1}^2 having at least one −2 coordinate. We prove that only 240 of them have a finite orbit, and solve 231 of them with our method. The 9 remaining models are the counterparts of the 4 models of the small step case that resist the uniform solution method (and which are known to have an algebraic generating function). We conjecture D-finiteness for their generating functions, but only two of them are likely to be algebraic. We also prove non-D-finiteness for the 12 870 models with an infinite orbit, except for 16 of them.

3. Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior by Stephen Melczer and Mark C. Wilson. SIAM Journal on Discrete Mathematics, Volume 33(4), 2140–2174.

Subsumes results in earlier conference paper (FPSAC 2016). We consider the enumeration of walks on the non-negative lattice $\mathbb{N}^d$, with steps defined by a set $S\subset\{âˆ’1,0,1\}^d - \{0\}$. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps $S$ is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions with non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets which are symmetric over all but one axis. As a special case, we apply our results when $d=2$ to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.

4. Quiver Asymptotics: $\mathcal{N}=1$ Free Chiral Ring by Sanjaye Ramgoolam, Mark C. Wilson and Ali Zahabi. Journal of Physics A: Mathematical and Theoretical 53 (10), 105401, 2020

The large $N$ generating functions for the counting of chiral operators in $\mathcal{N}=1$, four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with this asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine $\mathbb{C}^3/\tilde{A}_n$ orbifold quivers.

5. Asymptotic lattice path enumeration using diagonals by Stephen Melczer and Marni Mishna. Algorithmica 75, 782-811, 2016.

Uses the smooth point theory to deal with highly symmetric walks confined to the positive orthant.

6. Diagonal asymptotics for products of combinatorial classes by Mark C. Wilson. Combinatorics, Probability and Computing 24, 354-372, 2015 (Flajolet memorial issue).

We generalize and improve recent results by B\'{o}na and Knopfmacher and by Banderier and Hitczenko concerning the joint distribution of the sum and number of parts in tuples of restricted compositions. Specifically, we generalize the problem to general combinatorial classes and relax the requirement that the sizes of the compositions be equal. We extend the main explicit results to enumeration problems whose counting sequences are Riordan arrays. In this framework, we give an alternative method for computing asymptotics in the supercritical case, which avoids explicit diagonal extraction and seems likely to be computationally more efficient.

7. Quantum random walks on the integer lattice via generating functions by Andrew Bressler. PhD thesis, U. Penn., 2009.

This thesis contains several early results from the papers below. It also includes exact asymptotics for a three-chirality walk on the line and the 2-D Hadamard walk, as well as a proof of Airy phenomena for the two-chirality walk. It concludes with a preliminary discussion of higher dimensions.

8. Random walk on the integer lattice: examples and phenomena by Andrew Bressler, Torin Greenwood, Robin Pemantle, and Marko Petkovsek.

We apply results from the previous 2-D QRW paper to compute limiting probability profiles for various quantum random walks in one and two dimensions. Using analytic machinery we show some features of the limit distribution that are not evident in an empirical intensity plot of the time 10,000 distribution. Some conjectures are stated and computational techniques are discussed as well.

9. Two-dimensional quantum random walk by Yuliy Baryshnikov, Wil Brady, Andrew Bressler, and Robin Pemantle. Journal of Statistical Physics 142 (2011), 78-107.

We analyze several families of two-dimensional quantum random walks. The feasible region (the region where probabilities do not decay exponentially with time) grows linearly with time, as is the case with one-dimensional QRW. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. Our methods are based on analysis of the space-time generating function, following the methods of the first paper of Pemantle and Wilson. Some extensions to toral singularities are required.

10. Quantum random walks in one dimension via generating functions by Andrew Bressler and Robin Pemantle. Proceedings of International Conference on Analysis of Algorithms, Juan-les-Pins, 2007.

Extended abstract (full version to appear sometime).

11. Quantum random walks on $\mathbb{Z}^2$ by Wil Brady. MS thesis, U. Penn., 2007.

This thesis computes the asymptotic probability profile for nearest-neighbor quantum random walks in the two-dimensional integer lattice with various quantum coins (4x4 unitary matrices). A shape theorem is proved for the Hadamard coin (the standard choice). Analogous results for other coins, exhibiting an unexpectedly wide range of visual phenomena, are verified numerically but not proved. The numeric finite-time profiles are compared to pictures of the theoretically computed range of non-exponential decay. The shapes are indeed the same. Furthermore, plots of the normalized probabilities in the non-exponential region turn out to have internal structure that is mirrored in the rendering of the theoretically predicted non-exponential decay region via point plotting. This has led to followup work showing why both of these are essentially the same as a plot of the push-forward via the Gauss map of the surface measure on the toral part of the pole variety.

## Applications of this project by others

We try to update this a few times a year to give an idea of the variety of applications. There are many more citations, some of which use the methods, while others only mention them. A full list can be found using Google Scholar, for example. Citations to Pemantle-Wilson 2002 | Citations to 2008 SIAM Review paper | Citations to 2013 book
• Sample publications directly using the results of this project
• M. Kovacevic. Runlength-Limited Sequences and Shift-Correcting Codes: Asymptotic Analysis. IEEE Transactions on Information Theory 2019.
• T. George. Limit shapes for cube groves with periodic conductances. arxiv.org/abs/1711.00790.
• J. Courtiel, S. Melczer, M.Mishna, K. Raschel. Weighted lattice walks and universality criteria. Journal of Combinatorial Theory A 2017.
• R. Vidunas. Counting derangements and Nash equilibria. Annals of Combinatorics 2017.
• J. Pantone. The Asymptotic Number of Simple Singular Vector Tuples of a Cubical Tensor. arxiv.org/abs/1605.06099
• RF de Andrade, E Lundberg, B Nagle. Asymptotics of the extremal excedance set statistic. European Journal of Combinatorics 2015.
• A. Straub, W. Zudilin. Positivity of rational functions and their diagonals. Journal of Approximation Theory 2015.
• P Di Francesco, R Soto-Garrido. Arctic curves of the octahedron equation. Journal of Physics A 2014.
• I. Bena, M. Berkooz, J. de Boer, S. El-Showk, D. Scaling BPS solutions and pure-Higgs states. Journal of High Energy Physics 2012.
• Petersen, K. An adic dynamical system related to the Delannoy numbers. Ergodic Theory and Dynamical Systems 32 (2012), 809-823.
• M. Erickson, S. Fernando, K. Tran, Enumerating Rook and Queen Paths, Bulletin of the Institute for Combinatorics and Its Applications 60 (2010), 37--48.
• Noble, R. Asymptotics of a family of binomial sums. Journal of Number Theory 130 (2010), 2561-2585.
• Sahlmann, H. Entropy calculation for a toy black hole. Classical and Quantum Gravity, 200825 (5), art. no. 055004.
• Kong, Y. Asymptotics of the monomer-dimer model on two-dimensional semi-infinite lattices, Physical Review E 75, 051123 (2007) (13 pages).
• Kong, Y. Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density. Physical Review E 74, 061102 (2006) (15 pages).
• Corteel, S., Louchard, G., Pemantle, R. Common intervals in permutations. Discrete Mathematics and Theoretical Computer Science 8 (2006), pp. 189-214.
• Wormald, N. Tournaments with many Hamilton cycles. Preprint, 2001.

## Software

• Maple package and examples implementing code of Melczer and Salvy for smooth asymptotics.
• A Sage package by Alex Raichev has been incorporated into Sage from release 7.1 onward (some functionality may be broken in the latest Sage releases, and work is ongoing). The documentation is available.
• Old Maple worksheets dealing with examples in the "Twenty combinatorial examples ..." paper.

Last updated 2020-02-15.