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Anthony J Guttmann

Publications and source records attributed to Anthony J Guttmann.

At least 19 recordsLinked to original sources
arXiv2026

Counting LEGO configurations

We discuss the problem of counting certain LEGO structures, primarily those comprising parallel $w \times 1$ tiles. These can be combined, as a single LEGO structure, by interlocking the tiles. %Alternatively, if the interlocking condition is relaxed, so that tiles can also be placed end-to-end, a greater number of possible configurations results. We also study the historically earlier problem of counting the number of ways to combine $2 \times 4$ LEGO tiles, which in this case gives a 3-dimensional structure. In all cases the number of configurations is dominated by an exponential growth term, $\mu^n$ where $n$ is the number of tiles. We present an algorithm for counting these various LEGO configurations, and use the data to estimate the asymptotics. We analyse the data so generated, and conjecture that, for the two-dimensional structures, the number of possible configurations grows like $A(w)\mu(w)^n/n,$ and we give numerical estimates for $A(w)$ and $\mu(w)$ for $w < 11,$ while for the three-dimensional structure the number of possible configurations is conjectured to grow like $A\mu^n/n^{3/2},$ where $\mu = 117.25 \pm 0.05.$ We also study the sequences that arise when we fix the number of tiles $n,$ and vary the tile size $w.$ We prove that the sequences are polynomials of degree $n-1,$ and we give these explicitly for $n=1 \ldots 14.$

math.CO↗
arXiv2024

Counting occurrences of patterns in permutations

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes.

math.CO↗
arXiv2022

The gerrymander sequence, or A348456

Recently Kauers, Koutschan and Spahn announced a significant increase in the length of the so-called {\em gerrymander sequence}, given as A348456 in the OEIS, extending the sequence from 3 terms to 7 terms. We give a further extension to 11 terms, but more significantly prove that the coefficients grow as $λ^{4L^2},$ where $λ\approx 1.7445498, $ and is equal to the corresponding quantity for self-avoiding walks crossing a square (WCAS), or self-avoiding polygons crossing a square (PCAS). These are, respectively, OEIS sequences A007764 and A333323. Thus we have established a close connection between these previously separate problems. We have also related the sub-dominant behaviour to that of WCAS and PCAS, allowing us to conjecture that the coefficients of the gerrymander sequence A348456 grow as $λ^{4L^2+dL+e} \cdot L^g,$ where $d=-8.08708 \pm 0.0002,$ $e \approx 7.69$ and $g = 0.75 \pm 0.01,$ with $g$ almost certainly $3/4$ exactly. We also have generated 26 terms in the related gerrymander polynomial (defined below), and have been able to predict the asymptotic behaviour with a satisfying degree of precision. Indeed, it behaves exactly as $L$ times the corresponding coefficient of the generalised gerrymander sequence. The improved algorithm we give for counting these sequences is a variation of that which we recently developed for extending a number of sequences for SAWs and SAPs crossing a domain of the square or hexagonal lattices. It makes use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths.

math.CO↗
arXiv2022

Self-avoiding walks contained within a square

We have studied self-avoiding walks contained within an $L \times L$ square whose end-points can lie anywhere within, or on, the boundaries of the square. We prove that such walks behave, asymptotically, as walks crossing a square (WCAS), being those walks whose end-points lie at the south-east and north-west corners of the square. We provide numerical data, enumerating all such walks, and analyse the sequence of coefficients in order to estimate the asymptotic behaviour. We also studied a subset of these walks, those that must contain at least one edge on all four boundaries of the square. We provide compelling evidence that these two classes of walks grow identically. From our analysis we conjecture that the number of such walks $C_L$, for both problems, behaves as $$ C_L \sim λ^{L^2+bL+c}\cdot L^g,$$ where $λ= 1.7445498 \pm 0.0000012,$ $b=-0.04354 \pm 0.0005,$ $c=-1.35 \pm 0.45,$ and $g=3.9 \pm 0.1.$ Finally, we also studied the equivalent problem for self-avoiding polygons, also known as cycles in a square grid. The asymptotic behaviour of cycles has the same form as walks, but with different values of the parameters $c$, and $g$. Our numerical analysis shows that $λ$ and $b$ have the same values as for WCAS and that $c=1.776 \pm 0.002$ while $g=-0.500\pm 0.005$ and hence probably equals $-\frac12$.

math-ph↗
arXiv2022

Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices

We have analysed the recently extended series for the number of self-avoiding walks (SAWs) $C_L(1)$ that cross an $L \times L$ square between diagonally opposed corners. The number of such walks is known to grow as $λ_S^{L^2}.$ We have made more precise the estimate of $λ_S,$ based on additional series coefficients provided by several authors, and refined analysis techniques. We estimate that $λ_S = 1.7445498 \pm 0.0000012.$ We have also studied the subdominant behaviour, and conjecture that $$ C_L(1) \sim λ_S^{L^2+bL+c}\cdot L^g,$$ where $b=-0.04354 \pm 0.0001,$ $c=0.5624 \pm 0.0005,$ and $g=0.000 \pm 0.005.$ We implemented a very efficient algorithm for enumerating paths on the square and hexagonal lattices making use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths. Using this algorithm we extended and then analysed series for SAWs spanning the square lattice and self-avoiding polygons (SAPs) crossing the square lattice. These are known to also grow as $λ_S^{L^2}.$ The sub-dominant term $λ^b$ is found to be the same as for SAWs crossing the square, while the exponent $g = 1.75\pm 0.01$ for spanning SAWs and $g = -0.500 \pm 0.005$ for SAPs. We have also studied the analogous problems on the hexagonal lattice, and generated series for a number of geometries. In particular, we study SAWs and SAPs crossing rhomboidal, triangular and square domains on the hexagonal lattice, as well as SAWs spanning a rhombus. We estimate that the analogous growth constant $λ_H=1.38724951 \pm 0.00000005,$ so an even more precise estimate than found for the square lattice. We also give estimates of the sub-dominant terms.

math-ph↗
arXiv2021

Pattern-avoiding ascent sequences of length 3

Pattern-avoiding ascent sequences have recently been related to set-partition problems and stack-sorting problems. While the generating functions for several length-3 pattern-avoiding ascent sequences are known, those avoiding 000, 100, 110, 120 are not known. We have generated extensive series expansions for these four cases, and analysed them in order to conjecture the asymptotic behaviour. We provide polynomial time algorithms for the 000 and 110 cases, and exponential time algorithms for the 100 and 120 cases. We also describe how the 000 polynomial time algorithm was detected somewhat mechanically given an exponential time algorithm. For 120-avoiding ascent sequences we find that the generating function has stretched-exponential behaviour and prove that the growth constant is the same as that for 201-avoiding ascent sequences, which is known. The other three generating functions have zero radius of convergence, which we also prove. For 000-avoiding ascent sequences we give what we believe to be the exact growth constant. We give the conjectured asymptotic behaviour for all four cases.

math.CO↗
arXiv2020

Asymptotics of 3-stack-sortable permutations

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we conjecture that the generating function behaves as $$W(t) \sim C_0(1-μ_3 t)^α\cdot \log^β(1-μ_3 t), $$ so that $$[t^n]W(t)=w_n \sim \frac{c_0μ_3^n}{ n^{(α+1)}\cdot \log^λ{n}} ,$$ where $μ_3 = 9.69963634535(30),$ $α= 2.0 \pm 0.25.$ If $α= 2$ exactly, then $λ= -β+1$, and we estimate $β\approx -3.$ If $α$ is not an integer, then $λ=-β$, but we cannot give a useful estimate of $β$. The growth constant estimate (just) contradicts a conjecture of the first author that $$9.702 < μ_3 \le 9.704.$$ We also prove a new rigorous lower bound of $μ_3\geq 9.4854$, allowing us to disprove a conjecture of Bóna. We then further extend the series using differential-approximants to obtain approximate coefficients $O(t^{2000}),$ expected to be accurate to $20$ significant digits, and use the approximate coefficients to provide additional evidence supporting the results obtained from the exact coefficients.

math.CO↗
arXiv2020

Statistical Mechanics of Confined Polymer Networks

We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical $Θ$-point. In the $Θ$-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, $γ_b^Θ$, to that of terminally-attached arches, $γ_{11}^Θ,$ and to the correlation length exponent $ν^Θ.$ We find $γ_b^Θ = γ_{11}^Θ+ν^Θ.$ In the case of the special transition, we find $γ_b^Θ({\rm sp}) = \frac{1}{2}[γ_{11}^Θ({\rm sp})+γ_{11}^Θ]+ν^Θ.$ For general networks, the explicit expression of configurational exponents then naturally involve bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm-Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the case of ordinary, mixed and special surface transitions, and of the $Θ$-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.

math-ph↗
arXiv2019

Two-dimensional interacting self-avoiding walks: new estimates for critical temperatures and exponents

We investigate, by series methods, the behaviour of interacting self-avoiding walks (ISAWs) on the honeycomb lattice and on the square lattice. This is the first such investigation of ISAWs on the honeycomb lattice. We have generated data for ISAWs up to 75 steps on this lattice, and 55 steps on the square lattice. For the hexagonal lattice we find the $\theta$-point to be at $u_\mathrm{c} = 2.767 \pm 0.002.$ The honeycomb lattice is unique among the regular two-dimensional lattices in that the exact growth constant is known for non-interacting walks, and is $\sqrt{2+\sqrt{2}}$, while for half-plane walks interacting with a surface, the critical fugacity, again for the honeycomb lattice, is $1+\sqrt{2}$. We could not help but notice that $\sqrt{2+4\sqrt{2}} = 2.767\ldots .$ We discuss the difficulties of trying to prove, or disprove, this possibility. For square lattice ISAWs we find $u_\mathrm{c}=1.9474 \pm 0.001,$ which is consistent with the best Monte Carlo analysis. We also study bridges and terminally-attached walks (TAWs) on the square lattice at the $\theta$-point. We estimate the exponents to be $\gamma_b=0.00 \pm 0.03,$ and $\gamma_1=0.55 \pm 0.03$ respectively. The latter result is consistent with the prediction $\gamma_1(\theta) = \nu =\frac47$, albeit for a modified version of the problem, while the former estimate appears to be new.

cond-mat.stat-mech↗
arXiv2019

New scaling laws for self-avoiding walks: bridges and worms

We show how the theory of the critical behaviour of $d$-dimensional polymer networks gives a scaling relation for self-avoiding {\em bridges} that relates the critical exponent for bridges $γ_b$ to that of terminally-attached self-avoiding arches, $γ_{1,1},$ and the {correlation} length exponent $ν.$ We find $γ_b = γ_{1,1}+ν.$ We provide compelling numerical evidence for this result in both two- and three-dimensions. Another subset of SAWs, called {\em worms}, are defined as the subset of SAWs whose origin and end-point have the same $x$-coordinate. We give a scaling relation for the corresponding critical exponent $γ_w,$ which is $γ_w=γ-ν.$ This too is supported by enumerative results in the two-dimensional case.

cond-mat.stat-mech↗
arXiv2018

On consecutive pattern-avoiding permutations of length 4, 5 and beyond

We review and extend what is known about the generating functions for consecutive pattern-avoiding permutations of length 4, 5 and beyond, and their asymptotic behaviour. There are respectively, seven length-4 and twenty-five length-5 consecutive-Wilf classes. D-finite differential equations are known for the reciprocal of the exponential generating functions for four of the length-4 and eight of the length-5 classes. We give the solutions of some of these ODEs. An unsolved functional equation is known for one more class of length-4, length-5 and beyond. We give the solution of this functional equation, and use it to show that the solution is not D-finite. For three further length-5 c-Wilf classes we give recurrences for two and a differential-functional equation for a third. For a fourth class we find a new algebraic solution. We give a polynomial-time algorithm to generate the coefficients of the generating functions which is faster than existing algorithms, and use this to (a) calculate the asymptotics for all classes of length 4 and length 5 to significantly greater precision than previously, and (b) use these extended series to search, unsuccessfully, for D-finite solutions for the unsolved classes, leading us to conjecture that the solutions are not D-finite. We have also searched, unsuccessfully, for differentially algebraic solutions.

math.CO↗
arXiv2018

Counting Planar Eulerian Orientations

Inspired by the paper of Bonichon, Bousquet-Mélou, Dorbec and Pennarun, we give a system of functional equations which characterise the ordinary generating function, $U(x),$ for the number of planar Eulerian orientations counted by edges. We also characterise the ogf $A(x)$, for 4-valent planar Eulerian orientations counted by vertices in a similar way. The latter problem is equivalent to the 6-vertex problem on a random lattice, widely studied in mathematical physics. While unable to solve these functional equations, they immediately provide polynomial-time algorithms for computing the coefficients of the generating function. From these algorithms we have obtained 100 terms for $U(x)$ and 90 terms for $A(x).$ Analysis of these series suggests that they both behave as $const\cdot (1 - μx)/\log(1 - μx),$ where we conjecture that $μ= 4π$ for Eulerian orientations counted by edges and $μ=4\sqrt{3}π$ for 4-valent Eulerian orientations counted by vertices.

math.CO↗
arXiv2017

Numerical studies of Thompson's group F and related groups

We have developed polynomial-time algorithms to generate terms of the cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group, $(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We have also given an improved algorithm for the coefficients of Thompson's group $F,$ giving 32 terms of the cogrowth series. We develop numerical techniques to extract the asymptotics of these various cogrowth series. We present improved rigorous lower bounds on the growth-rate of the cogrowth series for Thompson's group $F$ using the method from \cite{HHR15} applied to our extended series. We also generalise their method by showing that it applies to loops on any locally finite graph. Unfortunately, lower bounds less than 16 do not help in determining amenability. Again for Thompson's group $F$ we prove that, if the group is amenable, there cannot be a sub-dominant stretched exponential term in the asymptotics\footnote{ }. Yet the numerical data provides compelling evidence for the presence of such a term. This observation suggests a potential path to a proof of non-amenability: If the universality class of the cogrowth sequence can be determined rigorously, it will likely prove non-amenability. We estimate the asymptotics of the cogrowth coefficients of $F$ to be $$ c_n \sim c \cdot \mu^n \cdot \kappa^{n^\sigma \log^\delta{n}} \cdot n^g,$$ where $\mu \approx 15,$ $\kappa \approx 1/e,$ $\sigma \approx 1/2,$ $\delta \approx 1/2,$ and $g \approx -1.$ The growth constant $\mu$ must be 16 for amenability. These two approaches, plus a third based on extrapolating lower bounds, support the conjecture \cite{ERvR15, HHR15} that the group is not amenable.

math.GR↗
arXiv2016

Permutations sortable by two stacks in series

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further coefficients approximately. Analysing these coefficients by a variety of methods we conclude that the OGF behaves as $$S(z) \sim A (1 - μ\cdot z)^γ,$$ where $μ=12.45 \pm 0.15,$ $γ= 1.5 \pm 0.3,$ and $A \approx 0.02$.

math.CO↗
arXiv2016

Series extension: Predicting approximate series coefficients from a finite number of exact coefficients

Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly well in predicting (approximately) subsequent coefficients. These can then be used by the ratio method to obtain improved estimates of critical parameters. In favourable cases, given only the first 20 coefficients, the next 100 coefficients are predicted with useful accuracy. More surprisingly, this is also the case when the method of differential approximants does not do a useful job in estimating the critical parameters, such as those cases in which one has stretched exponential asymptotic behaviour. Nevertheless, the coefficients are predicted with surprising accuracy. As one consequence, significant computer time can be saved in enumeration problems where several runs would normally be made, modulo different primes, and the coefficients constructed from their values modulo different primes. Another is in the checking of newly calculated coefficients. We believe that this concept of approximate series extension opens up a whole new chapter in the method of series analysis.

cond-mat.stat-mech↗
arXiv2014

On the growth rate of 1324-avoiding permutations

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in 5 further terms of the generating function. We analyse the known coefficients and find compelling evidence that unlike other classical length-4 pattern-avoiding permutations, the generating function in this case does not have an algebraic singularity. Rather, the number of 1324-avoiding permutations of length $n$ behaves as $$B\cdot μ^n \cdot μ_1^{n^σ} \cdot n^g.$$ We estimate $μ=11.60 \pm 0.01,$ $σ=1/2,$ $μ_1 = 0.0398 \pm 0.0010,$ $g = -1.1 \pm 0.2$ and $B =9.5 \pm 1.0.$

math.CO↗
arXiv2014

Analysis of series expansions for non-algebraic singularities

Existing methods of series analysis are largely designed to analyse the structure of algebraic singularities. Functions with such singularities have their $n^{th}$ coefficient behaving asymptotically as $A \cdot \mu^n \cdot n^g.$ Recently, a number of problems in statistical mechanics and combinatorics have been encountered in which the coefficients behave asymptotically as $B \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where typically $\sigma = \frac{1}{2}$ or $\frac{1}{3}.$ Identifying this behaviour, and then extracting estimates for the critical parameters $B, \,\, \mu, \,\, \mu_1, \,\, \sigma, \,\, {\rm and} \,\, g$ presents a significant numerical challenge. We describe methods developed to meet this challenge.

math-ph↗