Papers

Integral systems in real composition algebras give rise to finite metric configurations whose geometry is linked to both regular polytopes and root-systems. In this work we investigate, to our knowledge for the first time in this form, the shell polytopes obtained by fixing the integral norm and taking the convex hull of the corresponding integral elements. The first shells recover the familiar root-polytopal configurations attached to the classical Hurwitz systems, while the Okubo algebra gives a quite different behaviour. The Okubo integral closure does not recover the Gosset polytope directly: it selects a two-adic hierarchy whose first visible layers are a cross-polytope and a \(D_8\) root polytope. We further show that the natural intermediate lattice is isometric to the rescaled cubic lattice; consequently every shell decomposes into explicit orbits of the hyperoctahedral group \(W(B_8)\), and the higher Okubo shells admit a complete combinatorial description in cubic-lattice coordinates. The full \(E_8\) Gosset polytope is then recovered from the intermediate lattice by maximal-isotropic gluing along \((\ZZ/2)^4\). This gives an interplay between non-unital composition, integral lattice shadows, and the geometry of \(E_8\).

Suppose that $I$ is a unit square. Let $T$ (resp. $\Delta$) be an isosceles right triangle (resp. an equilateral triangle). We prove that any collection of triangles homothetic to $T$ (resp. $\Delta$), whose total area does not exceed $\frac{1}{2}$ (resp. $\frac{\sqrt{3}}{4}$), can be parallel packed into $I$. These upper bounds are tight.

A graph $G$ is said to be perfectly divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into two sets $A, B$ such that $H[A]$ is perfect and $\omega(B) < \omega(H)$. It is easy to see that the chromatic number of a perfectly divisible graph is at most $\binom{\omega(G)+1}{2}$. Hoàng conjectured that every graph $G$ with $\alpha(G) \le 3$ is perfectly divisible. We disprove this conjecture. In the same vein, a graph $G$ with at least one edge is $k$-divisible if for every induced subgraph $H$ of $G$ with at least one edge, the vertex set $V(H)$ can be partitioned into $k$ sets, none of which contains a largest clique of $H$. It is easy to see that the chromatic number of a $k$-divisible graph is at most $k^{\omega-1}$. Hoàng conjectured that every even-hole-free graph is 3-divisible. We confirm this conjecture.

We show that the realization theorem of Fernández de Bobadilla, which identifies the Milnor fiber of a weighted-homogeneous polynomial with the complement of a germ of analytic set, can be combined with the systematic Massey product constructions of Grbić-Linton for moment-angle complexes $\mathcal{Z}_K = \mathcal{Z}_K(D^2, S^1)$ to produce weighted-homogeneous polynomials whose Milnor fibers are arbitrarily highly connected and non-formal. The original application of this strategy, due to Fernández de Bobadilla, used the Denham-Suciu classification of lowest-degree triple Massey products and yielded only 2-connected non-formal Milnor fibers. The Grbić-Linton framework, which constructs non-trivial $n$-fold Massey products in $H^*(\mathcal{Z}_K;\mathbb{Q})$ for arbitrary $n$ and in arbitrary cohomological degrees, removes this connectivity restriction entirely.

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in\mathcal{G}\}$. In this note, we establish the product version of the Hilton-Milner Theorem for $k\geq 8$ in the full range. That is, if $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are non-trivial cross-intersecting, $n\geq 2k+1$ and $k\geq 8$, then \[ |\mathcal{F}||\mathcal{G}|\leq \left(\binom{n-1}{k-1}- \binom{n-k-1}{k-1} +1\right)^2. \]

We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph Laplacian. Next, by using the cyclic symmetry of $C_N^k$, we derive a spectral representation via Fourier analysis. Furthermore, by applying factorization and partial fraction decomposition of the corresponding difference operator, we obtain an explicit formula for the average hitting times consisting of a quadratic term and finitely many correction terms. These correction terms are described by second-order linear recurrence sequences associated with the characteristic polynomials, and can be regarded as natural generalizations of Fibonacci-type sequences. As a consequence, our formulas recover the known results for cycle graphs and squares of cycle graphs in a unified way. Moreover, from the formulas obtained for average hitting times, we derive explicit formulas for the effective resistances, the numbers of spanning trees, the numbers of two-component spanning forests, and the numbers of spanning trees of vertex-identified graphs. In particular, for the third power graph $C_N^3$ of the cycle graph, all of these quantities are written explicitly in terms of complex conjugate Fibonacci-type sequences. Our results clarify structural relations between random walk quantities and combinatorial quantities on cycle power graphs.

The classical no-three-in-line problem asks for the largest number $D(n)$ of points that can be chosen from an $n\times n$ grid with no three collinear; it remains open whether the elementary upper bound $2n$ is always attainable. We study a checkerboard-restricted variant in which all chosen points are monochromatic under the colouring of the grid by the parity of $x+y$. If $D_{\mathrm{mono}}(n)$ denotes the largest number of monochromatic points with no three collinear, the monochromatic diagonals already give $D_{\mathrm{mono}}(n)\le 2n-2$. The main object of the paper is a four-direction linear-programming relaxation on a fixed colour class, using rows, columns and the two diagonal families of slopes $\pm1$. For the ordinary square-grid problem this relaxation gives the trivial bound; on the checkerboard it is substantially tighter, and for $2\le n\le 16$ its floor agrees with the exact single-colour optimum except at four side lengths, where the gap is one. After symmetry reduction the dual relaxation has three one-dimensional reduced forms, according to the parity of $n$ and the chosen colour class. The central construction is an exact continuum dual certificate for the continuum problem associated with the scaled symmetry-reduced odd-fat case: explicit nonnegative functions $A$ and $B$ satisfying the continuum obstacle inequalities, with objective value the middle real root $\alpha\approx1.5768$ of $401\alpha^3-1744\alpha^2+2240\alpha-768=0$. Combined with a continuum-to-discrete sampling theorem of Aujla, Prellberg and Sandhu, this certificate yields the asymptotic upper bound $D_{\mathrm{mono}}(n)\le\alpha n+O(1)$. Finite LP computations are consistent with $\alpha$ as the exact limiting slope of the relaxation, and the exact small-$n$ data suggest, more speculatively, that the true checkerboard optimum tracks the same scale.

This paper proves four conjectured generating series, due to Chapoton, which concern invariants of posets and polytopes associated with a specific sequence of arbors. Two of these conjectures provide closed-form formulas for the generating series of the Zeta polynomial and the generating series of the M-triangle of the poset, respectively. The remaining two conjectures pertain, respectively, to the Ehrhart polynomial and the Laplace transform of the volume function of the associated arbor polytope.

In this paper, we consider circulant graphs obtained from the complete graph $K_N$ by deleting all edges belonging to a prescribed distance class. We study, in a unified manner, the effective resistance, the expected hitting time, the number of spanning trees, and the number of two-component spanning forests of these graphs. For general distance-class deletions, these quantities admit natural spectral representations in terms of the Laplacian eigenvalues. However, such representations typically remain at the level of finite Fourier sums, and concise closed forms are not expected in general. We focus on the case of a single deleted distance class. When the number of vertices $N$ is odd and $\gcd(r,N)=1$, the graph $G_{N,r}$ is isomorphic to $G_{N,1}$. In this setting, we derive explicit exponential-type formulas for the effective resistance and the number of spanning trees, and obtain corresponding closed expressions for two-component spanning forests and expected hitting times. Our results show that the case $r=2$ is not essentially new, but follows from a general isomorphism structure underlying distance-class deletions. We also clarify the relation of our formulas to earlier results on the complete graph with a Hamiltonian cycle removed, and provide a unified derivation within a spectral framework. Moreover, by asymptotic analysis, we show that the ratio $\tau(G_{N,1})/\tau(K_N)$ converges to $e^{-2}$ as $N \to \infty$.

We study the discrete graph-metric analogue of Gromov's filling area problem for the cycle graph \(C_n\). An abstract triangulation \(K\) is an isometric filling of \(C_n\) if \(\partial K=C_n\) and the graph distance between any two boundary vertices is not shortened inside the \(1\)-skeleton of \(K\). Let \(D(n;\epsilon)\) denote the minimum number of vertices in a \((1-\epsilon)\)-Lipschitz filling of \(C_n\), and set \[ D^*=\liminf_{\epsilon\to0^+}\liminf_{n\to\infty}\frac{D(n;\epsilon)}{n^2}. \] Previous work gives the general lower bound \(D^*\ge 1/8\), while discretizing the hemisphere gives the upper bound \[ D^*\le \frac{1}{\pi\sqrt3}. \] In this paper we give an explicit discrete construction which improves the hemispherical upper bound. More precisely, we construct isometric fillings \(K_n\) of \(C_n\) with \[ |V(K_n)|\le \left(\frac16+o(1)\right)n^2, \] and hence \[ D^*\le \frac16<\frac{1}{\pi\sqrt3}. \] This can directly illustrate the discrete filling area problem is a proper relaxation of Gromov's original filling area problem and cannot be used to settle Gromov's conjecture. The construction is a concentric annular filling.

We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $\chi$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(\alpha(H),\omega(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above: (1) every component of $T$ is a broom and $H$ is complete multipartite; or (2) $T$ is a forest and $H$ is complete bipartite. These two unify and substantially extend a number of previous results on linear and polynomial $\chi$-boundedness for various graph classes. For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions.

We consider the configuration model and the uniform simple graph with given degree sequence $\boldsymbol{d}=\left(d_i\right)_{i=1}^n$. We derive quantitative bounds for the errors in (i) joint normal-Poisson approximation to the numbers of isolated edges, isolated 2-stars, self-loops and double edges in the configuration model, and (ii) normal approximation to the numbers of isolated edges and isolated 2-stars conditioned on that the configuration model is simple. The latter provides the first finite sample normal approximation results for the uniform simple graph with given vertex degrees. To achieve this, we develop a new Stein's method for joint normal-Poisson approximation and a new coupling approach to sums of indicators, which may be of independent interest.

We introduce moonflowers, a weaker analogue of sunflowers. A family of sets $S_1,\ldots,S_k$ is a $k$-moonflower if each set $S_i$ contains at least one element that is absent from all the others. We study the extremal problem of determining the largest possible size of a family of sets of size at most $w$ that avoids a $k$-moonflower, and obtain near-optimal bounds. As an application, we revisit the code sparsification problem studied by Brakensiek and Guruswami (STOC 2025) and improve the bounds to near optimal. Concretely, we improve the dependence on the block length from poly-logarithmic to logarithmic, and show that such a dependence is necessary.

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric circulant, and they satisfy $AA^\top+BB^\top+CC^\top+DD^\top=4nI$. Using a computer search, we find all inequivalent near Williamson matrices for all odd orders at most $35$. We also show that such matrices exist for all odd orders up to $63$. As a consequence, we find the first known example of a quaternary Hadamard matrix of order $118$.

A finite generalized quadrangle $\cS$ is a Singer quadrangle if it has an automorphism group that acts sharply transitively on its points. In this paper, we introduce the notion of multipliers for a Singer quadrangle and study their basic properties. As an application, we show that a point-primitive automorphism group of a thick generalized quadrangle cannot have O'Nan-Scott type HS (holomorph simple), which answers an open problem in \cite{Bamberg 2019}.

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap F\neq \emptyset$ for all $F\in \mathcal{F}$. In 1980, the first author proved that for sufficiently large $n$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$. There was very little progress during more than forty years but recently (cf. \cite{FW25}) with a completely different approach we proved the same result for the full range $n\geq 2k$ and $k\geq 7$. In this short paper we prove the same inequality for all the remaining cases.

By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(\lambda)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the reverse semi-standard Young tableaux (resp. reverse marginally large tableaux), thereby providing a combinatorial description of the corresponding polyhedral realizations. Furthermore, a crystal structure on the set of Gelfand-Tsetlin patterns is obtained via the correspondence between the polyhedral realization of $\mathcal{B}(\lambda)$ and the reverse tableaux. As applications of our framework, we present concrete combinatorial realizations of the crystal embedding of $\mathcal B(\lambda)$ into $\mathcal B(\infty)$ and the set of Lusztig data.

We introduce and study a generalization of conformal rigidity for graphs. A graph is $k$-conformally rigid if the uniform edge weights simultaneously maximize the sum of the $k$ smallest nontrivial Laplacian eigenvalues and minimize the sum of the $k$ largest, over all normalized non-negative weight assignments. A graph that is $k$-conformally rigid for every $k$ is called totally conformally rigid. Our main result is a complete characterization: a graph is totally conformally rigid if and only if it is edge-rigid, meaning every canonical spectral embedding onto a Laplacian eigenspace is edge-isometric. We further show this is equivalent to all edges of the graph being pairwise Laplacian-cospectral, that is, the removal of any single edge yields a graph with the same Laplacian characteristic polynomial. Using semidefinite programming duality, we establish this equivalence and derive a polynomial-time algorithm for deciding edge-rigidity using only integer arithmetic. We provide a combinatorial characterization of edge-rigidity in terms of Laplacian walks and connect it to the walk-regularity of signed line graphs. We show that a graph is edge-rigid if and only if it is either $1$-walk-regular or $1$-walk-biregular, and we finally show an equivalence based on monotone gauges and gauge duality. As an application, we derive two non-trivial combinatorial consequences of total conformal rigidity, relating it to the number of spanning trees and the Kirchhoff index of the graph.

For the OEIS sequence A025166, defined by $a(n) = -n!\,2^{n}\,L_{n}(1/2)$ where $L_{n}$ is the Laguerre polynomial of degree $n$, R.~J.~Mathar contributed in February 2013 the conjectured order-2 P-recursive recurrence \[ a(n) + (-4n+3)\, a(n-1) + 4(n-1)^{2}\, a(n-2) \;=\; 0, \qquad n \ge 2. \] We give a one-page proof. The exponential generating function $F(x) = -\exp\!\big(-x/(1-2x)\big)/(1-2x)$ satisfies the first-order linear ODE $(1-2x)^{2} F'(x) = (1-4x)\, F(x)$, and Mathar's recurrence then falls out by reading off the coefficient of $x^{n}/n!$. Both steps are short. The supplementary archive includes a SymPy script which checks the ODE identically and the recurrence numerically up to $n = 5000$.

Let $\mathrm{Mac}(W)$ be the MacNeille completion of the Bruhat order of a Coxeter group $W$. We introduce an action of the $0$-Hecke monoid of type $W$ on $\mathrm{Mac}(W)$, which allows us to define a weak order and a descent set statistic on $\mathrm{Mac}(W)$. When $W$ is of type $A$, we recover constructions of Hamaker and Reiner, which were originally formulated in terms of monotone triangles and alternating sign matrices. Using this action, we prove that certain unions of Knutson–Miller subword complexes are vertex-decomposable. By specializing to type $A$, we prove a conjecture of Escobar, Klein, and Weigandt regarding Cohen–Macaulay ASM varieties. Along the way, we also exhibit a counterexample to a conjecture of Hamaker and Reiner regarding the poset topology of intervals in the ASM weak order. When $W$ is finite, our action also gives rise to a Boolean cell complex with one facet for each element of $\mathrm{Mac}(W)$. We prove that this is an abstract simplicial complex and that every linear extension of the weak order on $\mathrm{Mac}(W)$ is a shelling order for its facets. Finally, when $W$ is finite and irreducible, we use our $0$-Hecke action to introduce a noninvertible dynamical system on $\mathrm{Mac}(W)$ that we call the MacNeille pop-stack operator, and we prove that the maximum number of iterations of this operator needed to reach the bottom state is $h-1$, where $h$ is the Coxeter number of $W$. This article is meant to serve as a case study in using large language models to automate the workflow of mathematical research. The proof of the conjecture of Escobar–Klein–Weigandt and the disproof of the conjecture of Hamaker–Reiner were obtained autonomously by ChatGPT 5.4 Pro. Other aspects of the paper were obtained mostly by the author, but ChatGPT expedited the process. We provide a detailed account of this interaction.

We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs up to the exponential order. We do this by viewing a connected graph with a given degree distribution as the realization of the giant component in a larger configuration model, and carefully choosing the degree distribution of the larger graph so that it is likely that its giant component has the required degree distribution. To ensure that the connected graph has exactly the correct degrees, we use a switching argument. Additionally, we obtain results on rare event probabilities and describe the local structure of a uniform connected graph with a fixed degree sequence.

We prove a strengthening of Chang's lemma for subsets of $\mathbb F_p^n$. The classical conclusion that the large spectrum is contained in a subspace of dimension at most $2\varepsilon^{-2}\log(1/\alpha)$ is refined to show that every character outside this subspace has small correlation with the set not only globally, but also on average over the cosets of the orthogonal complement, in a natural cosetwise $\ell^1$ norm. As a consequence, we obtain a localized counting lemma. We also give an extension of the argument to arbitrary finite abelian groups.

An imprimitive symmetric indecomposable association scheme of rank $5$ is said to be Higmanian. In the present paper, we prove a necessary and sufficient condition for a Higmanian association scheme with two nontrivial parabolics to be uniform. We also provide examples of uniform Higmanian Cayley schemes.

A Tatra association scheme is an association scheme arising from a symmetric bilinear form defined on the equivalence classes of nonzero $2$-dimensional vectors modulo some subgroup of the multiplicative group of a finite field. In the present paper, we prove that every such association scheme is $2$-separable, i.e. it is determined up to isomorphism by the tensor of its $2$-dimensional intersection numbers.

In this paper, we study the asymptotic behaviour of the number of subtrees and the subtree density for a sequence of trees that converges in the Benjamini-Schramm sense. Benjamini-Schramm convergence, also called local weak convergence, describes the local behaviour of a sequence of graphs. Here we show that for a Benjamini-Schramm-convergent sequence of trees, the subtree entropy, i.e., the logarithm of the number of subtrees divided by the order, converges to a constant depending only on the limit. The same holds true for the subtree density, i.e., the probability of a uniformly random vertex being contained in a uniformly random subtree, provided that long paths are ruled out in the limit. Related to this, we show that the subtree density and the average subtree entropy are dense in different parts of the unit interval $[0,1]$ for both general trees and series-reduced trees.

Tree-child networks are an important class of phylogenetic network used to model reticulate evolutionary processes. These networks have attracted increasing attention from researchers with interests in both combinatorics and algorithms. A fundamental open problem posed by Pons and Batle asks whether the number $TC_{n,k}$ of bicombining tree-child networks with $n$ leaves and $k$ reticulation nodes equals the number of certain constrained words, now called Pons-Batle words. In this paper, we confirm the conjecture for tree-child networks with a bounded number of reticulation nodes. Our approach is combinatorial and analytic. We introduce families of Young tableaux with walls and holes and construct explicit bijections with Pons-Batle words, yielding a direct combinatorial explanation of the identities. These tableaux encode structural features of the underlying networks, including the placement of reticulation nodes. By projecting them to decorated Dyck paths, we obtain algebraic generating functions with differential operators encoding step weights, leading to explicit recurrence relations and closed-form formulas for $TC_{n,k}$. Beyond finite verification for moderate $k$, the framework reveals an underlying probabilistic structure. For $k=1$, natural structural parameters, such as the position and value of distinguished cells, converge, after rescaling, to $\mathrm{Beta}(2,1)$, $\mathrm{Beta}(1,2)$, and Uniform (i.e., $\mathrm{Beta}(1,1)$) distributions. These limit laws arise from a coalescence of singularities at the dominant square-root singularity, producing a non-analytic transition in the local expansion. Overall, our results provide both combinatorial insight and a unified analytic perspective on the asymptotic behavior of tree-child networks, showing how algebraic generating functions with interacting singularities systematically produce Beta limit laws.

We study the properties of the sequence of words $(B_i)$, where $B_1 = 101$ and $B_{i+1} = B_i C_i$ for $i \geq 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence ${\bf b} = 10101101011011101 \cdots$ of which all the $B_i$ are prefixes. We show that $\bf b$ is recurrent, but not uniformly recurrent; it has exponential factor complexity; it is not morphic; and the density of $1$'s exists and is transcendental.

Many papers have studied inequalities for Andrews and Paule's broken $k$-diamond partition function $\Delta_{k}(n)$ when $k=1$ or $2$. In this paper, we derive an exact formula for $\Delta_{k}(n)$ when $k\geq 1$. Building on this result, we also derive an asymptotic formula for $\Delta_{k}(n)$ with an explicit error bound. Using this formula, we prove that for $k\geq 1$ and sufficiently large $n$, $\Delta_{k}(n)$ satisfies the Turán and Laguerre inequalities of any order and exhibits asymptotic complete monotonicity. Define $n_k:=\max\left\{\left\lceil8k^{3}+\frac{k+1}{12}\right\rceil,526\right\}$. Furthermore, we show that $\Delta_{k}(n)$ is log-concave for $k\ge3$ and $n\ge n_k$. Consequently, it follows that $\Delta_{k}(a)\Delta_{k}(b)\ge\Delta_{k}(a+b)$ for $k\ge3$ and $a,b \ge n_k$.

We prove that every class of Eulerian directed graphs of bounded carving width (equivalently of bounded degree and treewidth) is well-quasi-ordered by strong immersion. In fact, we prove a stronger result, namely that every class of Eulerian directed graphs of bounded carving width, where every vertex is additionally labeled from a well-quasi-order, fixes a linear order on its incident edges, and may impose further restrictions on how the immersion is allowed to route paths through it, is well-quasi-ordered by an adequate notion of strong immersion. To this extent, we develop a framework seemingly suited to prove well-quasi-ordering for classes of Eulerian directed graphs by (strong) immersion and present a first meta theorem in that direction. We complement our results by observing that the class of Eulerian directed graphs of unbounded degree is not well-quasi-ordered by strong immersion, even if we assume the treewidth of the class to be at most two. We conclude with a dichotomy result, proving for a very restricted class of Eulerian directed graphs of unbounded degree that it is not well-quasi-ordered by strong immersion, but it is well-quasi-ordered by weak immersion.

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 $n=1 \ldots 14.$