Given a configuration $A$ of $n$ points in $\mathbb{R}^{d-1}$, we introduce the higher secondary polytopes $\Sigma_{A,1},\dots, \Sigma_{A,n-d}$, which have the property that $\Sigma_{A,1}$ agrees with the secondary polytope of Gelfand--Kapranov--Zelevinsky, while the Minkowski sum of these polytopes agrees with Billera--Sturmfels' fiber zonotope associated with (a lift of) $A$. In a special case when $d=3$, we refer to our polytopes as higher associahedra. They turn out to be related to the theory of total positivity, specifically, to certain combinatorial objects called plabic graphs, introduced by the second author in his study of the totally positive Grassmannian. We define a subclass of regular plabic graphs and show that they correspond to the vertices of the higher associahedron $\Sigma_{A,k}$, while square moves connecting them correspond to the edges of $\Sigma_{A,k}$. Finally we connect our polytopes to soliton graphs, the contour plots of soliton solutions to the KP equation, which were recently studied by Kodama and the third author. In particular, we confirm their conjecture that when the higher times evolve, soliton graphs change according to the moves for plabic graphs.

The Tutte polynomial is a well-studied invariant of graphs and matroids. We first extend the Tutte polynomial from graphs to hypergraphs, and more generally from matroids to polymatroids, as a two-variable polynomial. Our definition is related to previous works of Cameron and Fink and of Kálmán and Postnikov. We then define the universal Tutte polynomial Tn, which is a polynomial of degree n in 2+(2n−1) variables that specializes to the Tutte polynomials of all polymatroids (hence all matroids) on a ground set with n elements. The universal polynomial Tn admits three kinds of symmetries: translation invariance, Sn-invariance, and duality.

The reduced expressions for a given element w of a Coxeter group (W, S) can be regarded as the vertices of a directed graph R (w); its arcs correspond to the braid moves. Specifically, an arc goes from a reduced expression (a) over right arrow to a reduced expression (b) over right arrow when (b) over right arrow is obtained from (a) over right arrow by replacing a contiguous subword of the form stst. . . (for some distinct s, t is an element of S) by tsts. . . (where both subwords have length m(s,t), the order of st is an element of W). We prove a strong bipartiteness-type result for this graph R (w): Not only does every cycle of R (w) have even length; actually, the arcs of R (w) can be colored (with colors corresponding to the type of braid moves used), and to every color c corresponds an "opposite" color c(oP) (corresponding to the reverses of the braid moves with color c), and for any color c, the number of arcs in any given cycle of R (w) having color in {c, c(oP)}is even. This is a generalization and strengthening of a 2014 result by Bergeron, Ceballos and Labbe.

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We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with sorted sets, which earlier appeared in the context of alcoved polytopes and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the Eulerian number. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the weakly separated sets. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated cluster algebra.

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We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie, and Kenyon and Winkler to any hyperplane arrangement A. The volume of the resulting configuration space of connected branched polymers associated to the hyperplane arrangement A is expressed through the value of the characteristic polynomial of A at 0. We give a more general definition of the space of branched polymers, where we do not require connectivity, and introduce the notion of q-volume for it, which is expressed through the value of the characteristic polynomial of A at -q. Finally, we relate the volume of the space of branched polymers to broken circuits and show that the cohomology ring of the space of branched polymers is isomorphic to the Orlik-Solomon algebra.

We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines.We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings.Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the lattice points of a zonotope.

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincare polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincare polynomial. Our main technical tools are chordal graphs and perfect elimination orderings.

In 1826 N. Abel found a generalization of the binomial formula. In 1902 Abel’s theorem was further generalized by A. Hurwitz. In this paper we describe constructions that provide infinitely many identities each being a generalization of a Hurwitz’s identity. Moreover, we give combinatorial interpretations of all these identities as the forest volumes of certain directed graphs.

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on the cells, and describe how they are glued to each other.

The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhance a result of Fulton and Woodward by showing that the minimal monomial in the quantum parameters that occurs in the quantum product of two Schubert classes has a simple interpretation in terms of directed paths in this graph. We define path Schubert polynomials, which are quantum cohomology analogs of skew Schubert polynomials recently introduced by Lenart and Sottile. They are given by sums over paths in the quantum Bruhat graph of type A. The 3-point Gromov-Witten invariants for the flag manifold are expressed in terms of these polynomials. This construction gives a combinatorial description for the set of all monomials in the quantum parameters that occur in the quantum product of two Schubert classes.

This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the Gromov-Witten invariants of the Grassmannian with respect to the groups S3, (Z/nZ)2, and Z/2Z. The last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q . Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of q with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius invariants of graphic and cographic arrangements in terms of Hermite polynomials.

Recently, Berenstein et al. have proposed a duality between a sector of = 4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit considered, the effective 't Hooft coupling has been argued to be ?' = gYM2N/J2 = 1/(?p+?')2. We study Yang-Mills theory at small ?' (large ?) with a view to reproducing string interactions. We demonstrate that the effective genus counting parameter of the Yang-Mills theory is g22 = J4/N2 = (4?gs)2(?p+?')4, the effective two-dimensional Newton constant for strings propagating on the pp-wave background. We identify g2(?')1/2 as the effective coupling between a wide class of excited string states on the pp-wave background. We compute the anomalous dimensions of BMN operators at first order in g22 and ?' and interpret our result as the genus one mass renormalization of the corresponding string state. We postulate a relation between the three-string vertex function and the gauge theory three-point function and compare our proposal to string field theory. We utilize this proposal, together with quantum mechanical perturbation theory, to recompute the genus one energy shift of string states, and find precise agreement with our gauge theory computation.

In this paper, we give a combinatorial proof via lattice paths of the following result due to Andrews and Bressoud: for t⩽1, the number of partitions of n with all successive ranks at least t is equal to the number of partitions of n with no part of size 2−t. The identity is a special case of a more general theorem proved by Andrews and Bressoud using a sieve.

The group (Z/nZ)^2 is shown to act on the Gromov-Witten invariants of the complex flag manifold. We also deduce several corollaries of this result.

The tree volume of a weighted graph $G$ is the ``sum'''' of the tree volumes of all spanning trees of $G$, and the tree volume of a weighted tree $T$ is the product of the edge weights of $T$ times the ``product'''' of the letters of the Prüfer code of $T$ where the vertices of $G$ are viewed as independent indeterminants that can be multiplied and commute. The forest volume of $G$ is the tree volume of the graph $G^c$ obtained from $G$ by adding a new vertex $c$ and connecting every vertex of $G$ with $c$ by an arc of weight 1. We show that the forest volume is a natural generalization of the Laplacian polynomial of graphs and that it also can be expressed as the characteristic polynomial of a certain matrix similar to the Laplacian matrix. It turns out that the forest volumes of graphs possesses many important properties of the Laplacian polynomials, for example, the reciprocity theorem holds also for the forest volumes. We describe two constructions of graph compositions, and show that the forest volume of a composition can be easily found if the ``structure'''' of the composition and the forest volumes of the graph--bricks are known. As an illustration of the results on the forest volume of graph--compositions we give a combinatorial interpretation and proof of Hurwitz''s identity. Keywords: graph, tree, forest spanning tree, Laplacian matrix and polynomial, tree and forest volume.