Papers

Cizma and Linial asked for a classification of the metrizable theta graphs. We solve both their problem and its strict analogue. For $a\le b\le c$, the theta graph $\Theta_{a,b,c}$ is metrizable if and only if $a\le 2$ or $(a,b,c)=(3,3,3)$, and it is strictly metrizable if and only if $a\le2$. Thus $\Theta_{3,3,3}$ is precisely the exceptional theta graph that is metrizable but not strictly metrizable. The negative directions follow from the known obstructions $\Theta_{3,3,4}$ and $\Theta_{3,3,3}$ together with topological-minor closure. The positive direction is constructive. Consistency turns the possible detours through a length-two arm into compatible Ferrers relations, which are represented by one-dimensional potentials; all resulting shortest-path comparisons have positive slack. The exceptional ordinary-metrizable graph $\Theta_{3,3,3}$ is handled by a two-threshold weak Ferrers representation. The proof is structural, yields rational edge lengths algorithmically, and uses no enumeration of path systems.

Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. For the Glauber dynamics associated with the Gibbs measure defined by this Hamiltonian at any inverse temperature $\beta>0$, we show that, in any dimension $d\ge 2$, the infinite volume relaxation time satisfies \[e^{\beta^2/C}/C \le T_{\mathrm{rel}}\le Ce^{e^{C\beta}}\] for some $C>0$. Our result entails that the Gibbs measure is unique. The $e^{\beta^2}$ scaling was conjectured by Newman and Moore in 1999 and matches the behaviour of supercritical rooted kinetically constrained models such as the East model, thus recovering fragile glass phenomenology in the absence of kinetic constraints. More precisely, we show that, on a torus of side length $2^k$, when $\beta\to\infty$ and $k/\beta\to0$, we have $T_{\mathrm{rel}}=e^{2\beta k(1+o(1))}$. Quite surprisingly, however, we also prove that, on non-periodic finite domains of size $n\le e^{\beta/C}$ for large $C>0$, we have the much larger asymptotics $\ln T_{\mathrm{rel}}=\beta n^{\Theta(1)}$. The main ingredients of the proofs are new results in extremal and enumerative combinatorics and rely on renormalisation ideas for the dynamics and its groundstates also known as the Ledrappier subshift. We note consequences of our results to geometric group theory (more precisely to the complexity of the word problem for the Baumslag finitely presented group) and to ergodic theory.

In his 2009 paper, Thakur posed three conjectural hypotheses for the degrees of the power sums \[ S_d(k)=\sum_{\substack{a\in \mathbb F_q[t] \text{ monic}\\ \deg a=d}} a^{-k} \qquad\text{and}\qquad s_d(k)=-\deg_t S_d(k). \] For prime fields $q=p$, we prove Hypotheses H1 and H2, giving a unique greedy description of the extremal term in Carlitz's formula and establishing the recursion \[ s_d(k)=s_{d-1}(s_1(k))+s_1(k). \] As consequences, the prime-field recursion gives the strict Newton-polygon convexity used in the prime-field Carlitz-Goss Riemann-hypothesis theorem, and it recovers Thakur's nonvanishing theorem for positive multizeta values over $\mathbb F_p[t]$. We also prove Hypothesis H3 for all finite fields $q=p^f$, establishing the monotonicity \[ s_d(k)<s_d(k+1)\qquad (p\nmid k). \] We provide Lean formalizations of the arguments in this paper, generated by AxiomProver.

We study balanced subfamilies of the middle layer $\binom{[2n]}{n}$ of the Boolean lattice $2^{[2n]}$. A family $\mathcal{F}\subseteq\binom{[2n]}{n}$ is said to be balanced if every element in $[2n]$ appears in the same number of members of $\mathcal{F}$. A balanced subfamily of size 2 is exactly a complementary pair $\{A,[2n]\setminus A\}$, and therefore a family with no balanced subfamily of size $2$ has at most $\frac{1}{2}\binom{2n}{n}$ members. We show that for every $k\geq 1$ and all sufficiently large $n$, this maximum size is compatible with delaying the smallest size of a balanced subfamily until $2k+2$. More precisely, there exists a family $\mathcal{F}\subseteq\binom{[2n]}{n}$ of size $\frac{1}{2}\binom{2n}{n}$ with no balanced subfamilies of sizes $2,4,\ldots,2k$, but with a balanced subfamily of size $2k+2$. The proof is constructive and uses a general lifting from simple games to self-dual selectors in the middle layer. This provides an efficient constructive proof of a recent conjecture of Moss and Pedersen.

A classical theorem of Sheehan in 1977 states that every hamiltonian graph $G$ of order $n$ satisfying $e(G)>\left\lfloor \frac{n^2}{4}\right\rfloor+1$ contains at least two cycles of every length $\ell$, $3\le \ell\le n$. In the same paper, Sheehan recorded a conjecture of Hilton, which strengthens this conclusion by asserting that such a graph contains at least $n-\ell+2$ cycles of length $\ell$ for each $3\le \ell\le n$. We prove Hilton's conjecture for all hamiltonian graphs of order at least $440$.

For every $d \geq 4$, we construct a family of $(d-1)$-dimensional flag simplicial spheres $\mathcal K_n$ whose graphs contain independent sets of size asymptotically equal to the number of vertices. More precisely, we prove that for sufficiently large $n$, $$ \alpha(G(\mathcal K_n)) \geq f_0(\mathcal K_n) - \frac{C\,f_0(\mathcal K_n)}{\left(\log f_0(\mathcal K_n)\right)^{\lfloor d/2 \rfloor-1}},$$ where $C = C(d) > 0$. This disproves a recent conjecture of Chudnovsky and Nevo.

We consider the class of acyclic connected directed graphs with $N\geq 1$. In this paper we find the optimal upper bound for the number of paths amongst acyclic, connected graphs with $N$ edges. We prove that it is in fact optimal by finding an acyclic, connected graph with $N$ edges that realizes this bound. We then adapt these methods to find an optimal bound for Leavitt path algebras over a finite, acyclic, connected graph with $N$ edges.

In this paper, we introduce a new idea to study modules of logarithmic differential forms of hyperplane arrangements, which we call residue ideals. We first establish basic properties of these ideals, including their radicals and primary decompositions, and obtain applications for freeness of restrictions of arrangements. Then we apply these ideals to the study of modules of logarithmic differential $1$-forms for graphic arrangements. We give an explicit generating set for these modules and find a new connection to cover ideals of graphs studied in combinatorial commutative algebra. As a consequence we establish several new connections between arrangement theory and Stanley–Reisner theory.

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients. In this paper, we study the Schur positivity of symmetric functions naturally associated with set partitions, with respect to two different notions of descent. In the first case, the Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings. In the second case, the Schur functions correspond to two-rows Young diagrams, and the coefficients are partial sums of associated Bell numbers. A key ingredient of our approach in the second case is the notion of a removable singleton, defined algebraically and shown to admit an equivalent combinatorial interpretation via jeu-de-taquin rectification of skew tableaux. As an application, we establish Schur positivity for various classes of symmetric functions indexed by non-crossing partitions and partitions with a given number of parts. We provide an explicit combinatorial description of the tableaux that contribute to the Schur expansion, and we connects the obtained coefficients to some known integer sequences.

This paper proves that every $2$-connected regular matroid of sufficiently large rank has a $2$-connected graphic flat of large rank. Furthermore, we explicitly determine a list of such unavoidable flats.

In the first version of Chang, Slote, Volberg, and Zhang's paper \cite{BSA_of_PTF}, the authors modify a nice recursive approach due to Kane in \cite{Correct_exponent_for_AS} where he bounded the average sensitivity of polynomial threshold functions. In \cite{BSA_of_PTF} Kane's argument was adopted to estimate the boolean surface area of polynomial threshold function. The bridge is a combinatorial averaging lemma considering all balanced partitions. The lemma serves as a substitute for an additive property of average sensitivity. With the lemma, one can apply a Kane-type algorithm to derive a recurrence. Solving the recurrence then gives an upper bound of $e^{C_d \sqrt{\log n}}$ for the boolean surface area. In the second version of the same paper, the authors derive a polylog upper bound for BSA of PTFs. The difference is that they use a tail estimate for the sensitivity function. With the help of a polynomial restriction lemma in \cite{poly_restriction} they sharpen the upper bound. It is noteworthy that when applying the polynomial restriction, each coordinate is put into each part independently with equal probability. As a result, a partition does not necessarily have equal-size blocks. In other words, it may not be balanced. In this note, we first investigate the effect of different partitioning. Second, we use the recursive method in the first version to derive a polylog upper bound for $\mathbb E[s(x)^{\eta}]$ where $\eta < 1/2$. It is interesting to note the phase transition that happens at $\eta=1/2$ in both versions of the proof (but in a completely different form). Section 2.5 treats that.

One way to construct Difference Distance Magic Oriented Graphs (DDMOGs) is via a recently introduced technique called weighted sum. We explore the quality of said construction further by introducing the notion of an ornament. An ornament is an oriented graph that, when used in a weighted sum with an existing DDMOG, creates a new DDMOG. We provide results on the construction of a specific type of ornaments, called s-nodes ornaments. We conclude the paper with a list of open questions related to DDMOGs and ornaments.

We study the problem of enumerating symmetric chain decompositions (SCDs) of the minuscule lattices $L(m,n)$ of partitions in an $m$ by $n$ box and $M(n)$ of partitions into distinct parts at most $n$. We shift the focus from constructing a single SCD to analyzing the global structure of the set of SCDs. Let $\#SCD(P)$ be the number of symmetric chain decompositions of $P$. We give an explicit formula for $\#SCD(L(2,n))$ based on inversion sets of permutations and conjecture that for fixed $m>1$ both $\#SCD(L(m,n))$ and $\#SCD(M(n))$ grow super-exponentially. These conjectures are supported by data produced by AlphaEvolve, an evolutionary coding agent from Google DeepMind, and are in the same vein as a recent paper of Tomon on the growth rate of $\#SCD$ for the Boolean lattice and hypergrid. We make connections with crystal bases and show that the Lusztig involution (evacuation) extends to an involution on SCDs, which we use to show that $\#SCD(M(n))$ is even for $n>2$. We use skew tableaux sequences, which are equivalent to SCDs, and describe a potential path forward for finding SCDs through a notion of tableaux avoidance. We discuss implications of the conjectures for the problem of computing plethysm coefficients and discuss connections to physics and geometry. We end with a list of conjectures, questions and open problems.

Azuma et al. showed that a cactus-expandable Boolean network $\Sigma$ is structurally oscillatory if all the simple cycles of $\Sigma$ contain an odd number of inhibiting edges. We show that we do not need the cactus-expandable condition. Namely, a strongly connected Boolean network $\Sigma$ is structurally oscillatory if and only if all the simple cycles of $\Sigma$ contain an odd number of inhibiting edges. Additionally, we provide a characterization of a structurally oscillatory Boolean network for the general case.

Let $S = (D, \sigma)$ be a signed digraph, where $D=(\mathcal{V},\mathcal{A})$ is the underlying digraph of $S$ and $\sigma:\mathcal{A}\rightarrow\{-1,+1\}$ is a sign function. In this paper, we study the adjacency singular values of signed digraphs. We provide a necessary and sufficient condition for the singular values of a digraph to remain invariant under signing. We study rank of signed digraphs and determine signed digraphs with rank one. As an application of this result, we obtain a lower bound for the trace norm of signed digraphs and characterize the extremal signed digraphs.

Let $H$ be a 3-partite 3-uniform hypergraph whose three vertex classes all have size $n$. For a vertex $v \in V(H)$, the link graph $N_H(v)$ is defined on $V(H)\setminus\{v\}$ with edge set $\{e\setminus\{v\}: v\in e\in E(H)\}$, and we denote by $\rho(N_H(v))$ its spectral radius. We prove that for every $\alpha>0$ there exists $n_0$ such that for all $n\ge n_0$ the following holds: if \[ \rho\bigl(N_H(v)\bigr) > \left(\frac{\sqrt{2}}{2}+\alpha\right)n \] for every vertex $v\in V(H)$, then $H$ contains a perfect matching. This spectral condition is asymptotically best possible.

The saturation number $\mu^*(G)$ of a graph $G$ is the minimum cardinality of a maximal matching, and $H(G)$ is its harmonic index. TxGraffiti conjectured in 2023 that $\mu^*(G) \le H(G)$ for every nontrivial connected graph $G$, and B{\i}y{\i}koğlu refuted this by showing that the ratio $\mu^*(G)/H(G)$ can be made arbitrarily large. Restricting to trees bounds the ratio sharply. Every nontrivial tree $T$ satisfies $\mu^*(T) < \frac{3}{2} H(T)$, with the constant $3/2$ best possible. A complementary bound $H(G) < 4\mu^*(G)$ holds for every graph with an edge, so on a nontrivial tree the saturation number is pinned to $\frac{1}{4} H(T) < \mu^*(T) < \frac{3}{2} H(T)$, both constants best possible. The friendship graph $F_4$ is a smallest counterexample to the conjecture, on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices. For each positive integer $m$ a family of graphs with $m$ hubs has ratio approaching $m+1$, while the conjecture holds whenever all vertices have equal degree. Both invariants arise in applications, the harmonic index as a molecular descriptor and the saturation number as a measure of adsorption inefficiency, and the bounds estimate the latter, which is NP-hard to compute, by the former, which is computable in linear time.

We prove the existence of clopen strong marker sets in $F(2^{\mathbb{Z}^n})$ for arbitrary finite generating sets. Specifically, for any positive integers $n, d_0\geq 1$ and any finite generating set $S\subseteq \mathbb{Z}^n$, we construct a clopen set $M\subseteq F(2^{\mathbb{Z}^n})$ and a positive integer $\Delta$ such that (1) for any distinct $x,y\in M$ in the same orbit, $\rho(x,y)\geq d_0$; (2) for any $v\in S$ and any $x\in F(2^{\mathbb{Z}^n})$, there are non-negative integers $a,b\leq \Delta$ such that $av\cdot x\in M$ and $-bv\cdot x\in M$. Here $\rho$ denotes the Euclidean metric. The same result then holds for the standard supremum-norm metric $\rho_\infty$ (with an adjusted constant), by the equivalence of norms on $\mathbb{Z}^n$. The proof introduces polyhedral packages in $\mathbb{R}^n$ as a generalization of the rectangular packages used in earlier work, enabling the construction to handle generating vectors with arbitrary coordinate patterns. As an application, we obtain a continuous proper edge $(2|S|+1)$-coloring of the Schreier graph on $F(2^{\mathbb{Z}^n})$ with generating set $S$, recovering a result of Gao–Wang–Wang.

We prove a sharp product theorem for $r$-cross-intersecting families in the $p$-biased measure. If $r\ge2$, $0\le p\le \frac{r-1}{r}$, and $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq 2^{[n]}$ are $r$-cross-intersecting, then $$\prod_{i=1}^r \mu_p(\mathcal{F}_i)\le p^r.$$ The bound is attained by a common $1$-star, and the range of $p$ is best possible. In particular, this proves the equal-bias case of a conjecture of Frankl and Tokushige and, for $r=3$, confirms a conjecture of Tokushige. We also prove a stability theorem: for $r\ge3$, every near-extremal $r$-tuple is close, in $p$-biased measure, to a common $1$-star, with an optimal linear dependence on the product deficit. The extremal proof uses a coordinatewise coupling at the critical bias together with an isoperimetric inequality for increasing families. The stability proof uses dual families and random ordered partitions to obtain Fourier concentration, then applies biased Friedgut–Kalai–Naor theorem to force the star structure.

An even factor of a graph $G$ is a spanning subgraph in which every vertex has positive even degree. It is known that the minimum degree $\delta(G)\ge 2$ is a trivial necessary condition for $G$ to have an even factor. Recent spectral results for the existence of even factors used the certain complete-join graphs as exceptional extremal graphs. However, these graphs already contain $2$-factors and therefore are not genuine obstructions. This observation leads to the natural problem of determining the true sharp spectral threshold when the minimum degree is given. In this paper, we provide tight adjacency spectral radius conditions for a connected graph to contain an even factor, and characterize all extremal graphs, respectively. We also study the stronger requirement of a connected even factor, equivalently a spanning connected Eulerian subgraph. For this property, we also establish the corresponding sharp adjacency spectral radius condition and determine the unique extremal graph.

For every integer $g\ge 2$, we determine exactly which integers occur in the greedy 3-sumfree sequence that starts with $1$, $g$, and $g+1$. This gives a direct proof of a conjecture of Bosma, Bruin, Fokkink, Grube, Reuijl, and Tromp. We also obtain an explicit eventual periodic description, including both the preperiod and the repeating block.

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The inclusion graph of annihilators in $R$, denoted by $\Gamma^{\prime}(R)$, is a graph with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}_R(x)\subseteq \operatorname{ann}_R(y)$ or $\operatorname{ann}_R(y)\subseteq \operatorname{ann}_R(x)$. It is proved that $\Gamma^{\prime}(R)$ is not connected if and only if $R$ is reduced with $|\operatorname{Min}(R)|=2$. Also, we show that if $\Gamma^{\prime}(R)$ is a connected graph, then the diameter of $\Gamma^{\prime}(R)$ is at most $4$ and the girth of $\Gamma^{\prime}(R)$ is at most $6$, if it contains a cycle. Moreover, we study the affinity between inclusion graph of annihilators and complement of the annihilator graph (a well-known graph with the same vertices and two distinct vertices $x$ and $y$ are adjacent if and only if $\operatorname{ann}_R(xy)\neq \operatorname{ann}_R(x)\cup \operatorname{ann}_R(y)$) associated with a commutative ring. Finally, we characterize all rings whose inclusion graphs of annihilators are complete.

We study labeling-sensitive Fourier complexity for finite graph kernels. After identifying the vertices of a graph with the cyclic group $\mathbb Z_N$, its adjacency matrix becomes a function on $\mathbb Z_N^2$. Minimizing the quotient of the $\ell^1$ and $\ell^2$ norms of its two-dimensional Fourier transform over all vertex labelings gives an isomorphism invariant $\operatorname{FR}_{\min}(G)$. A nuclear-norm argument gives \[\operatorname{FR}_{\min}(G) \geq \frac{\mathcal E(G)}{\sqrt{2s}},\] where $s$ is the number of edges and $\mathcal E(G)$ is the graph energy. The natural cyclic labeling attains equality for every circulant graph. We obtain exact formulas for several graph families and a labeling-sensitive complete bipartite example. We also connect the invariant with the Fourier algebra of $\mathbb Z_N^2$. The quantitative Cohen idempotent theorem implies that every Boolean kernel of bounded Fourier ratio has an exact signed coset decomposition whose length is independent of $N$. A previously established Fourier-ratio recovery theorem gives stable Frobenius approximation of a fixed labeled adjacency matrix from Bernoulli samples. We distinguish this conclusion from exact edge recovery and from the problem of finding a good labeling. For a Laplacian eigenvalue of multiplicity $m(\lambda)$, we prove \[ \operatorname{FR}_{\min}(\Pi_\lambda) \geq \sqrt{m(\lambda)}, \] with equality for circulant graphs. Strongly regular graphs and the Petersen graph show how adjacency and projector complexity can agree or differ. Direct projector sampling yields heat-kernel approximation. We conclude with a graph-signal spectral synthesis principle and asymptotic uniqueness from incomplete vertex data.

We study Motzkin paths whose up-, level-, and down-step weights are affine functions of the height: alpha_k = A k + alpha_0, gamma_k = B k + gamma_0, beta_k = C k + beta_0. Let w_{n,k} be the total weight of paths of length n ending at height k, and let W_n = w_{n,0} be the return column. For affine weights the row recurrence becomes a first-order differential equation in the variable marking terminal height, and, unless beta_0 = C, that equation involves the unknown return series as boundary data. We show that the return series is nevertheless determined by the step rule alone: a characteristic ending on the floor forces a cancellation and yields an Abel-Volterra equation for the returns, while the same identity read at an interior point reconstructs the entire triangle (w_{n,k}). We call this boundary cancellation the differential kernel method; it takes the place of substituting an admissible kernel root, which is unavailable because the equation is differential rather than algebraic in the catalytic variable. When the boundary index nu = (beta_0 - C)/C is a positive integer m, the construction becomes finite and combinatorial: shifting every height by m turns the divided-difference evolution into an ordinary weighted path model on a half-line carrying m virtual levels below the visible floor, and a first-entry decomposition expresses (w_{n,k}) through two such local models. Exactly one bridge, of weight alpha_0 - A, leads from the virtual strip back to the visible region; it is closed, and the terminal-height columns factorise, precisely when alpha_0 = A. For the Dyck weights alpha_k = k+1, beta_k = k+nu+1, gamma_k = 0 this gives sum_{n>=0} w_{n,k} t^n / n! = sec^{nu+1}(t) tan^k(t).

We give an exact finite-dimensional Perron–Frobenius realization of the dominant zero of the Nekrasov–Okounkov polynomials $\nop _n(z)$. For a normalized positive sequence $h=(h(n))_{n\ge 1}$ with $h(1)=1$, define $\pol _0^h(z)=1$ and, for $n\ge 1$, \[ \pol _n^h(z)=\frac{z}{h(n)}\sum_{k=1}^n \sigma(k)\pol _{n-k}^h(z),\] where $\sigma(k)$ denotes the sum of divisors of $k$. The Nekrasov–Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift $\nop _n(z)=\pol _n^h(z+1)$. We derive a Hessenberg determinant representation for $\pol _n^h(z)$. After separating the trivial zero at the origin, the remaining zeros of $\pol _n^h(-z)$ are identified with the eigenvalues of an explicit $(n-1)\times(n-1)$ nonnegative matrix $M_n^h$. We prove that $M_n^h$ is primitive and apply Perron–Frobenius theory to show that $\pol _n^h(z)$ has a unique zero of maximal modulus; this zero is real, negative, and simple. As a consequence, the same property holds for the Nekrasov–Okounkov polynomials. We also prove strict monotonicity of the associated spectral radii.

Sidorenko's conjecture asserts that for every bipartite graph $H$ and every graph $G$, \[ t(H,G)\geq t(K_2,G)^{e(H)}. \] A result of Szegedy shows that, in order to prove the conjecture, it suffices to verify the corresponding inequality on a special family of highly symmetric bipartite Cayley type hosts arising from symmetric groups. Motivated by this reduction, we study Cayley type bipartite kernels associated with functions on finite groups and their conjugacy class averages. Our first result gives a reduction through conjugacy averaging: for a fixed bipartite graph $H$, if the $H$-density of each Cayley type host is at least the $H$-density of its conjugacy class average, then $H$ is strong Sidorenko, and hence Sidorenko. Our second result proves a Sidorenko-type inequality for 1-subdivision graphs on conjugacy-averaged Cayley kernels associated with arbitrary real-valued functions on finite groups.

We study the distance from the root to the nearest leaf, the analogous quantity for a uniformly chosen vertex, and its protection number, in size-conditioned simply generated trees. We prove a uniform exponential tail bound for each of these quantities, valid for arbitrary offspring distributions. As a consequence, these random variables are uniformly integrable of every order. This yields convergence of all moments to those of the corresponding local limit. The argument is probabilistic and unified across the three quantities.

We study collapsibility in the multiparametric models of random simplicial complexes, namely the lower and upper models. In the upper model, we improve upon a result of Farber and Nowik, and assert that the homology is a.a.s concentrated in a single dimension by proving that the complex collapses to that \di. In the lower model, we prove that the complex a.a.s collapses to the \di\ with maximal non-trivial cohomology. We then compare this threshold to the ones derived previously for the special cases of the clique complex (by Kahle) and the Linial-Meshulam model.

In this paper, we study the family of subcomplexes of the order complexes of finite graded posets, defined via its rank function. We address three main topics. (1) We describe the general topological properties of these subcomplexes in relation to magnitude homology of graded posets. (2) For posets whose order complexes are simplicial subdivisions of closed manifolds, we show that the homology groups of these subcomplexes agree with that of the undelying manifold except for the top dimension, where it is a nontrivial free abelian group. (3) For shellable graded posets, we prove that each of the subcomplexes are also shellable. Moreover, in the case of geometric semilattices, we show that each subcomplexes are homotopy equivalent to a nontrivial wedge sums of spheres of the same dimension.

We prove that the Opdam-Cherednik kernel, also known as the nonsymmetric Opdam hypergeometric function, can be written as the Laplace transform of a positive measure supported on the convex hull of the Weyl group orbit of its argument. As a consequence, the trigonometric Dunkl intertwining operator is positivity preserving. The main ingredient in the proof is a new formula for the Opdam-Cherednik kernel as a degeneration of nonsymmetric Macdonald polynomials. As a further application, we prove majorization inequalities for Macdonald polynomials and Heckman-Opdam hypergeometric functions associated with arbitrary root systems.