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We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a $01$-matrix with many ones and without large all-one submatrices, and we put our results together and construct an $n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov rank $n-o(n)$., Comment: 9 pages

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to Sudakov and Tomon, and our proof builds up on their work. A recent construction of B. Janzer implies that our bound is tight up to an $O((\log n)^4 \log \log n)$ factor., Comment: 9 pages; corrected typos

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{\"a}ger \cite{shmuck} and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in \cite{Dyer}. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2102.10134

We introduce toric promotion as a cyclic analogue of Sch\"utzenberger's promotion operator. Toric promotion acts on the set of labelings of a graph $G$. We discuss connections between toric promotion and previously-studied notions such as toric posets and friends-and-strangers graphs. Our main theorem provides a surprisingly simple description of the orbit structure of toric promotion when $G$ is a forest., Comment: 10 pages, 5 figures, to be published in Proceedings of the AMS

A recent conjecture of Di Francesco states that the number of domino tilings of a certain family of regions on the square lattice is given by a product formula reminiscent of the one giving the number of alternating sign matrices. These regions, denoted ${\mathcal T}_n$, are obtained by starting with a square of side-length $2n$, cutting it in two along a diagonal by a zigzag path with step length two, and gluing to one of the resulting regions half of an Aztec diamond of order $n-1$. Inspired by the regions ${\mathcal T}_n$, we construct a family $C_{m,n}^{a,b,c,d}$ of cruciform regions generalizing the Aztec diamonds and we prove that their number of domino tilings is given by a simple product formula. Since (as it follows from our results) the number of domino tilings of ${\mathcal T}_n$ is a divisor of the number of tilings of the cruciform region $C_{2n-1,2n-1}^{n-1,n,n,n-2}$, the special case of our formula corresponding to the latter can be viewed as partial progress towards proving Di Francesco's conjecture., 14 pages

Given a fixed hypergraph H H , let wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) denote the smallest number of edges in an n n -vertex hypergraph G G , with the property that one can sequentially add the edges missing from G G , so that whenever an edge is added, a new copy of H H is created. The study of wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H H very little is known regarding wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) , Alon proved in 1985 that for every graph H H there is a limiting constant C H C_H so that wsat ⁡ ( n , H ) = ( C H + o ( 1 ) ) n \operatorname {wsat}(n,H)=(C_H+o(1))n . Tuza conjectured in 1992 that Alon’s theorem can be (appropriately) extended to arbitrary r r -uniform hypergraphs. In this paper we prove this conjecture.

For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summation $\sum_{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400]., 13 pages. This is a preliminary manuscript

The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics., Comment: Second version, some errors corrected, less pages, more results

The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the multilinear component of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\sum_{n\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations. We determine the plethystic inverse of the alternating sum of the odd Lie characteristics., Comment: 12 pages; Section 3 reorganised per referee comments; added Lemma 3.2 and Proposition 3.9. To appear in Proceedings of the American Mathematical Society

We study the diameter of the graph $G(w)$ of reduced words of an element $w$ in a Coxeter group $W$ whose edges correspond to applications of the Coxeter relations. We resolve conjectures of Reiner--Roichman and Dahlberg--Kim by proving a tight lower bound on this diameter when $W=S_n$ is the symmetric group and by characterizing the equality cases. We also give partial results in other classical types which illustrate the limits of current techniques., 14 pages, comments welcome

Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n \times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp(-c(n\log n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $\exp(-c n^{1/2})$ on the singularity probability, our method is different and considerably simpler., 12 pages

We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by k k for various cardinals k k . We show that for a probability measure preserving (p.m.p) graph G G , a measurable orientation can be found when k k is larger than the normalized cost of the restriction of G G to any positive measure subset. Using an idea of Conley and Tamuz, we can also find Borel orientations of graphs with subexponential growth; however, for every k k we also find graphs which admit measurable orientations with outdegree bounded by k k but no such Borel orientations. Finally, for special values of k k we bound the projective complexity of Borel k k -orientability for graphs and graphings of equivalence relations. It follows from these bounds that the set of equivalence relations admitting a Borel selector is Σ 2 1 \mathbf {\Sigma }_{2}^{1} in the codes, in stark contrast to the case of smooth relations.

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of products of either charged or neutral fermionic creation and annihilation operators, Wick's theorem and a factorization identity for VEV's of products of two mutually anticommuting sets of neutral fermionic operators., 17 pages. Typos corrected. Title word order changed. References updated

Let $\mathcal{S}_n$ be the set of all $n$-by-$n$ symmetric real matrices, and let $\mathcal{C}_n$ be the copositive cone, that is, the set of all matrices $a\in\mathcal{S}_n$ that fulfill the condition $u^\top a u\geqslant0$ for all $n$-vectors $u$ with nonnegative entries. We prove that a linear mapping $\varphi:\mathcal{S}_n\to \mathcal{S}_n$ satisfies $\varphi(\mathcal{C}_n)=\mathcal{C}_n$ if and only if $$\varphi(x)=m^\top xm$$ for a fixed monomial matrix $m$ with nonnegative entries., Comment: 5 pages

A recent breakthrough of Conlon and Ferber yielded an exponential improvement on the lower bounds for multicolor diagonal Ramsey numbers. In this note, we modify their construction and obtain improved bounds for more than three colors., 4 pages; minor edits from v1

In this note we generalise some of the work of Klarner on free semigroups of affine maps acting on the real line by using a classical approach from geometric group theory (the Ping-Pong lemma). We also investigate the boundaries within which Klarner's necessary condition for a semigroup to be related is applicable., 7 pages, 1 figure

For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this paper we prove that the dimension of the above projective eigenvariety is zero, i.e. there are finite many eigenvectors associated with the spectral radius up to a scalar. For a general nonnegative tensor, we characterize the nonnegative combinatorially symmetric tensor for which the dimension of projective eigenvariety associated with spectral radius is greater than zero. Finally we apply those results to the adjacency tensors of uniform hypergraphs., Comment: arXiv admin note: text overlap with arXiv:1707.07414

Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map $\theta\colon \mathbb{C}^n\to \mathbb{C}^n$ which outputs the critical values of the canonical polynomial constructed from the input, and they proved that this map is onto. Along the way, they showed that $\theta$ is a local homeomorphism whenever the entries of the input are distinct and nonzero, and, implicitly, they produced a polynomial expression for the Jacobian determinant of $\theta$. In this article we extend and generalize both the local homeomorphism result and the elegant determinant identity to analogous situations where the critical points occur with multiplicities. This involves stratifying $\mathbb{C}^n$ according to which coordinates are equal and generalizing $\theta$ to a similar map $\mathbb{C}^\ell \to \mathbb{C}^\ell$ where $\ell$ is the number of distinct critical points. The more complicated determinant identity that we establish is closely connected to the multinomial identity known as Dyson's conjecture., Comment: 13 pages

Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case., v2: Minor edits. To appear in Proceedings of the American Mathematical Society. 15 pages, 2 figures, 3 tables, comments welcome

Recently, Glasner, Tsankov, Weiss, and Zucker showed that if $\Gamma$ is an infinite discrete group, then every minimal $\Gamma$-flow is disjoint from the Bernoulli shift $2^\Gamma$. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lov\'{a}sz Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems., Comment: 5 pages; v2: minor changes following referee's comments

We prove an infinite-dimensional version of an approximate Ramsey theorem of Gowers, initially used to show that every Lipschitz function on the unit sphere of $c_0$ is oscillation stable. To do so, we use the theory of ultra-Ramsey spaces developed by Todorcevic in order to obtain an Ellentuck-type theorem for the space of all infinite block sequences in $\mathrm{FIN}_{\pm k}$., Comment: 15 pages; fixed typos, clarified notation and added references

We prove that the cone over a Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This was previously shown for four other classes of arrangements. We exhibit an infinite family of cones over Dirichlet arrangements that are combinatorially distinct from these other four classes., 10 pages, 4 figures

We prove a central limit theorem for the joint distribution of $s_q(A_jn)$, $1\le j \le d$, where $s_q$ denotes the sum-of-digits function in base~$q$ and the $A_j$'s are positive integers relatively prime to $q$. We do this in fact within the framework of quasi-additive functions. As application, we show that most elements of "Catalan-like" sequences - by which we mean integer sequences defined by products/quotients of factorials - are divisible by any given positive integer., Comment: AmS-LaTeX, 11 pages; minor corrections and improvements

We are interested in the asymptotics of the number of standard Young tableaux $f^{\lambda/\mu}$ of a given skew shape $\lambda/\mu$. We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth regimes of $|\mu|$ compared to $|\lambda|$, from $|\mu|$ fixed to $|\mu|$ of order $|\lambda|$. When $|\mu|=o(|\lambda|^{1/3})$, we get an asymptotic expansion to any order. When $|\mu|=o(|\lambda|^{1/2})$, we get a sharp upper bound. For bigger $|\mu|$, we prove a weaker bound and give a conjecture on what we believe to be the correct order of magnitude. Our results are obtained by expressing $f^{\lambda/\mu}$ in terms of irreducible character values of the symmetric group and applying known upper bounds on characters., Comment: 15 pages, v3: final version incorporating referee comments

We obtain a new $q$-analogue of the classical Leibniz series $\sum_{k=0}^\infty(-1)^k/(2k+1)=\pi/4$, namely \begin{equation*} \sum_{k=0}^\infty\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\frac{(q^2;q^2)_{\infty}(q^8;q^8)_{\infty}}{(q;q^2)_{\infty}(q^4;q^8)_{\infty}}, \end{equation*} where $q$ is a complex number with $|q, Comment: 11 pages

An $n\times n$ sign pattern $S$, which is a matrix with entries $0,+,-$, is called spectrally arbitrary if any monic real polynomial of degree $n$ can be realized as a characteristic polynomial of a matrix obtained by replacing the non-zero elements of $S$ by numbers of the corresponding signs. A sign pattern $S$ is said to be a superpattern of those matrices that can be obtained from $S$ by replacing some of the non-zero entries by zeros. We develop a new technique that allows us to prove spectral arbitrariness of sign patterns for which the previously known "Nilpotent Jacobian" method does not work. Our approach leads us to solutions of numerous open problems known in the literature. In particular, we provide an example of a sign pattern $S$ and its superpattern $S'$ such that $S$ is spectrally arbitrary but $S'$ is not, disproving a conjecture proposed in 2000 by Drew, Johnson, Olesky, and van den Driessche., 5 pages

We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known results for fixed $d\geq 3$ and for $d=n-1$, confirming a prediction of Nachmias and Peres on a question of Benjamini. As a corollary, for the largest component of the percolated random $d$-regular graph, we also determine the diameter and the mixing time of the lazy random walk. In contrast to previous approaches, our proof is based on a simple application of the switching method., Comment: 10 pages

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most $3$, and $F_4$ and we give computational evidence for $H_4$. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meeting $X$ from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups., 15 pages, results considerably strengthened, completely rewritten; v3: results strengthened, final version, accepted to Proceedings AMS

It is well known that strongly minimal groups are commutative. Whether this is true for various generalisations of strong minimality has been asked in several different settings (see Hyttinen [2002], Maesono [2007], Pillay and Tanovi\'c [2011]). In this note we show that the answer is positive for groups with locally modular homogeneous pregeometries.

Two finitely generated monoids are constructed, one finitely presented, the other not, whose (directed, unlabelled) Cayley graphs are isomorphic. Postprint

We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $ n$ vertices with maximum degree $ d$, showing that an independent set drawn uniformly at random from such a graph has expected size at least $ (1+o_d(1)) \frac {\log d}{d}n$. This gives an alternative proof of Shearer's upper bound on the Ramsey number $ R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $ d$ is at least $ \exp \left [\left (\frac {1}{2}+o_d(1) \right ) \frac {\log ^2 d}{d}n \right ]$. The constant $ 1/2$ in the exponent is best possible. In both cases, tightness is exhibited by a random $ d$-regular graph. Both results come from considering the hard-core model from statistical physics: a random independent set $ I$ drawn from a graph with probability proportional to $ \lambda ^{\vert I\vert}$, for a fugacity parameter $ \lambda >0$. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree $ d$. The bound is asymptotically tight in $ d$ for all $ \lambda =O_d(1)$. We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory

We prove an extension of Bourgain's theorem on pinned distances in measurable subset of $\mathbb{R}^2$ of positive upper density, namely Theorem $1^\prime$ in [Bourgain, 1986], to pinned non-degenerate $k$-dimensional simplices in measurable subset of $\mathbb{R}^{d}$ of positive upper density whenever $d\geq k+2$ and $k$ is any positive integer., Minor revision made. To appear in Proc. Amer. Math. Soc

We prove that, provided $d > k$, every sufficiently large subset of $\mathbf{F}_q^d$ contains an isometric copy of every $k$-simplex that avoids spanning a nontrivial self-orthogonal subspace. We obtain comparable results for simplices exhibiting self-orthogonal behavior., 11 pages; improved and revised

A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$ given by the Erd\H os-Ko-Rado Theorem. In particular, this extends the Hilton-Milner theorem on nontrivial intersecting families and answers a recent question of Han and Kohayakawa for large $n$. In the case $r=3$ we describe the structure of all intersecting families with more than 10 edges. We also prove a stability result for the Erdos matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977.

We show that some hyperbolic 3-manifolds which are tessellated by copies of the regular ideal hyperbolic tetrahedron embed geodesically in a complete, finite volume, hyperbolic 4-manifold. This allows us to prove that the complement of the figure-eight knot geometrically bounds a complete, finite volume hyperbolic 4-manifold. This the first example of geometrically bounding hyperbolic knot complement and, amongst known examples of geometrically bounding manifolds, the one with the smallest volume., 9 pages, 4 figures, typos corrected, improved exposition of tetrahedral manifolds. Added Proposition 3.3, which gives necessary and sufficient conditions for M_T to be a manifold, and Remark 4.4, which shows that the figure-eight knot bounds a 4-manifold of minimal volume. Updated bibliography

We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$., Comment: 12 pages, 1 figure

The deck, $\mathcal{D}(X)$, of a topological space $X$ is the set $\mathcal{D}(X)=\{[X \setminus \{x\}]\colon x \in X\}$, where $[Y]$ denotes the homeomorphism class of $Y$. A space $X$ is (topologically) reconstructible if whenever $\mathcal{D}(Z)=\mathcal{D}(X)$ then $Z$ is homeomorphic to $X$. It is known that every (metrizable) continuum is reconstructible, whereas the Cantor set is non-reconstructible. The main result of this paper characterises the non-reconstructible compact metrizable spaces as precisely those where for each point $x$ there is a sequence $\langle B_n^x \colon n \in \mathbb{N}\rangle$ of pairwise disjoint clopen subsets converging to $x$ such that $B_n^x$ and $B_n^y$ are homeomorphic for each $n$, and all $x$ and $y$. In a non-reconstructible compact metrizable space the set of $1$-point components forms a dense $G_\delta$. For $h$-homogeneous spaces, this condition is sufficient for non-reconstruction. A wide variety of spaces with a dense $G_\delta$ set of $1$-point components are presented, some reconstructible and others not reconstructible., Comment: 15 pages, 2 figures

We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis., Comment: 8 pages, to appear in Proceedings of the AMS

In a recent paper, Chernikov and Starchenko prove that graphs defined in distal theories have strong regularity properties, generalizing previous results about graphs defined by semi-algebraic relations. We give a shorter, purely model-theoretic proof of this fact., 6 pages

Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and most fundamental examples of a cluster algebra is the homogenous coordinate ring of the Grassmannian. We show that the Grassmannian is locally acyclic. Morally, we are in fact showing the stronger result that all positroid varieties are locally acyclic. However, it has not been shown that all positroid varieties have cluster structure, so what we actually prove is that certain cluster varieties associated to Postnikov's alternating strand diagrams are locally acylic. Moreover, we actually establish a slightly stronger property than local acyclicity, which we term the Louise property, that is designed to facilitate proofs involving the Mayer-Vietores sequence., Comment: 14 pages, 8 figures, minor edits from previous version, added references to recent work of LeClerc

We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac., 13 pages, revised; to appear in Proc. AMS

Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable and available to the research practitioner in these areas. Also, the methods of proof utilized here are simple and amenable to many more analogous computations. We conclude the paper with a list of such formulas., Comment: 13 pages, no figures

We prove a regularity lemma with respect to arbitrary Keisler measures mu on V, nu on W where the bipartite graph (V,W,R) is definable in a saturated structure M and the formula R(x,y) is stable. The proof is rather quick and uses local stability theory. The special case where (V,W,R) is pseudofinite, mu, nu are the counting measures and M is suitably chosen (for example a nonstandard model of set theory), yields the stable regularity theorem of Malliaris-Shelah (Transactions AMS, 366, 2014, 1551-1585), though without explicit bounds or equitability., 6 pages. This second version takes into account some comments of Sergei Starchenko that additional cases need to be handled in the proof of Lemma 2.1

We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i \leq m$ in such a way that the points $\{x_1,...,x_m\}\subset \mathbb{R}^d$ are in general position. We give two proofs, one elementary proof requiring slightly stronger conditions, and one proof using topological techniques in the spirit of Aharoni and Haxell's celebrated generalization of Hall's theorem.

We introduce to the context of multigraded modules the methods of modules over categories from algebraic topology and homotopy theory. We develop the basic theory quite generally, with a view toward future applications to a wide class of graded modules over graded rings. The main application in this paper is to study the Betti poset B=B(I,k) of a monomial ideal I in the polynomial ring R=k[x_1,...,x_m] over a field k, which consists of all degrees in Z^m of the homogeneous basis elements of the free modules in the minimal free Z^m-graded resolution of I over R. We show that the order simplicial complex of B supports a free resolution of I over R. We give a formula for the Betti numbers of I in terms of Betti numbers of open intervals of B, and we show that the isomorphism class of B completely determines the structure of the minimal free resolution of I, thus generalizing with new proofs results of Gasharov, Peeva, and Welker. We also characterize the finite posets that are Betti posets of a monomial ideal., To appear in Proceedings of the American Mathematical Society. Changed title and reorganized exposition significantly; results unchanged

We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles, generalizing Gysin formulas for Schur S- and Q-functions., Comment: 7 pages; the version corresponding to the published one and its corrigenda, and corrigenda published electronically on March 16, 2016

We give a bijection between permutations of length 2n and certain pairs of Dyck paths with labels on the down steps. The bijection arises from a game in which two players alternate selecting from a set of 2n items: the permutation encodes the players' preference ordering of the items, and the Dyck paths encode the order in which items are selected under optimal play. We enumerate permutations by certain statistics, AA inversions and BB inversions, which have natural interpretations in terms of the game. We give new proofs of classical identities such as \sum_p \prod_{i=1}^n q^{h_i -1} [h_i]_q = [1]_q [3]_q ... [2n-1]_q where the sum is over all Dyck paths p of length 2n, and the h_i are the heights of the down steps of p., 9 pages, 1 figure

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994) proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact probability of a random graph being planar near the critical point $M=n/2$. For each $\lambda$, we find an exact analytic expression for $$ p(\lambda) = \lim_{n \to \infty} \Pr{G(n,\textstyle{n\over 2}(1+\lambda n^{-1/3})) is planar}.$$ In particular, we obtain $p(0) \approx 0.99780$. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of $G(n,\textstyle{n\over 2})$ being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point., Comment: 10 pages, 1 figure

We prove an Alexander-type duality for valuations for certain subcomplexes in the boundary of polyhedra. These strengthen and simplify results of Stanley (1974) and Miller-Reiner (2005). We give a generalization of Brion's theorem for this relative situation and we discuss the topology of the possible subcomplexes for which the duality relation holds., 11 pages

We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix, and thus attract a special attention., 11 pages