arXiv subjects
Publications and source records attributed to Karola Meszaros.
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.
Given a matrix Schubert variety $\overline{X_π}$, it can be written as $\overline{X_π}=Y_π\times \mathbb{C}^q$ (where $q$ is maximal possible). We characterize when $Y_π$ is toric (with respect to a $(\mathbb{C}^*)^{2n-1}$-action) and study the associated polytope $Φ(\mathbb{P}(Y_π))$ of its projectivization. We construct regular triangulations of $Φ(\mathbb{P}(Y_π))$ which we show are geometric realizations of a family of subword complexes. Subword complexes were introduced by Knutson and Miller in 2004, who also showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.
Greene's rational function $Ψ_P({\bf x})$ is a sum of certain rational functions in ${\bf x}=(x_1, \ldots, x_n)$ over the linear extensions of the poset $P$ (which has $n$ elements), which he introduced in his study of the Murnaghan-Nakayama formula for the characters of the symmetric group. In recent work Boussicault, Féray, Lascoux and Reiner showed that $Ψ_P({\bf x})$ equals a valuation on a cone and calculated $Ψ_P({\bf x})$ for several posets this way. In this paper we give an expression for $Ψ_P({\bf x})$ for any poset $P$. We obtain such a formula using dissections of root polytopes. Moreover, we use the subdivision algebra of root polytopes to show that in certain instances $Ψ_P({\bf x})$ can be expressed as a product formula, thus giving a compact alternative proof of Greene's original result and its generalizations.
We study multiplication of any Schubert polynomial $\mathfrak{S}_w$ by a Schur polynomial $s_λ$ (the Schubert polynomial of a Grassmannian permutation) and the expansion of this product in the ring of Schubert polynomials. We derive explicit nonnegative combinatorial expressions for the expansion coefficients for certain special partitions $λ$, including hooks and the 2x2 box. We also prove combinatorially the existence of such nonnegative expansion when the Young diagram of $λ$ is a hook plus a box at the (2,2) corner. We achieve this by evaluating Schubert polynomials at the Dunkl elements of the Fomin-Kirillov algebra and proving special cases of the nonnegativity conjecture of Fomin and Kirillov. This approach works in the more general setup of the (small) quantum cohomology ring of the complex flag manifold and the corresponding (3-point) Gromov-Witten invariants. We provide an algebro-combinatorial proof of the nonnegativity of the Gromov-Witten invariants in these cases, and present combinatorial expressions for these coefficients.
Cover-inclusive Dyck tilings are tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths, in which tiles are no larger than the tiles they cover. These tilings arise in the study of certain statistical physics models and also Kazhdan--Lusztig polynomials. We give two bijections between cover-inclusive Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the second bijection maps the "discrepancy" between the upper and lower boundary of the tiling to descents of the linear extension.
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.
We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.
We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula $\prod_{k=1}^{n-2} Cat(k)$ for the type $A_n$ version, where $Cat(k)$ is the $k$th Catalan number, we introduce type $C_{n+1}$ and $D_{n+1}$ Chan-Robbins-Yuen polytopes along with intriguing conjectures pertaining to their properties.
Intrigued by the product formula prod_{i=1}^{n-2} C_i for the volume of the Chan-Robbins-Yuen polytope CRY_n, where C_i is the ith Catalan number, we construct a family of polytopes P_{m,n}, whose volumes are given by the product \prod_{i=m+1}^{m+n-2}\frac{1}{2i+1}{{m+n+i} \choose {2i}}. The Chan-Robbins-Yuen polytope CRY_n coincides with P_{0,n-1}. Our construction of the polytopes P_{m,n} is an application of a systematic method we develop for expressing volumes of a class of flow polytopes as the number of certain triangular arrays. This method can also be used as a heuristic technique for constructing polytopes with combinatorial volumes. As an illustration of this we construct polytopes whose volumes equal the number of r-ary trees on n internal nodes, \frac{1}{(r-1)n+1} {{rn} \choose n}. Using triangular arrays we also express the volumes of flow polytopes as constant terms of formal Laurent series.
The number of regions of the type A_{n-1} Shi arrangement in R^n is counted by the intrinsically beautiful formula (n+1)^{n-1}. First proved by Shi, this result motivated Pak and Stanley as well as Athanasiadis and Linusson to provide bijective proofs. We give a description of the Athanasiadis-Linusson bijection and generalize it to a bijection between the regions of the type C_n Shi arrangement in R^n and sequences a_1a_2...a_n, where a_i \in \{-n, -n+1,..., -1, 0, 1,..., n-1, n\}, i \in [n]. Our bijections naturally restrict to bijections between regions of the arrangements with a certain number of ceilings (or floors) and sequences with a given number of distinct elements. A special family of posets, whose antichains encode the regions of the arrangements, play a central role in our approach.
We study a family of identities regarding a divisibility property of the Kostant partition function which first appeared in a paper of Baldoni and Vergne. To prove the identities, Baldoni and Vergne used techniques of residues and called the resulting divisibility property "mysterious." We prove these identities entirely combinatorially and provide a natural explanation of why the divisibility occurs. We also point out several ways to generalize the identities.
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.
The type A_n full root polytope is the convex hull in R^{n+1} of the origin and the points e_i-e_j for 1<= i x_{ik}x_{ij}+x_{jk}x_{ik}+βx_{ik}, can be interpreted as triangulations of P(T). Using these triangulations, the volume and Ehrhart polynomial of P(T) are obtained. If we allow variables x_{ij} and x_{kl} to commute only when i, j, k, l are distinct, then the reduced form of m[T] is unique and yields a canonical triangulation of P(T) in which each simplex corresponds to a noncrossing alternating forest. Most generally, the reduced forms of all monomials in the noncommutative case are unique.
The type C_n full root polytope is the convex hull in R^n of the origin and the points e_i-e_j, e_i+e_j, 2e_k for 1 <= i < j <= n, k \in [n]. Given a graph G, with edges labeled positive or negative, associate to each edge e of G a vector v(e) which is e_i-e_j if e=(i, j), i < j, is labeled negative and e_i+e_j if it is labeled positive. For such a signed graph G, the associated root polytope P(G) is the intersection of the full root polytope with the cone generated by the vectors v(e), for edges e in G. The reduced forms of a certain monomial m[G] in commuting variables x_{ij}, y_{ij}, z_k under reductions derived from the relations of a bracket algebra of type C_n, can be interpreted as triangulations of P(G). Using these triangulations, the volume of P(G) can be calculated. If we allow variables to commute only when all their indices are distinct, then we prove that the reduced form of m[G], for "good" graphs G, is unique and yields a canonical triangulation of P(G) in which each simplex corresponds to a noncrossing alternating graph in a type C sense. A special case of our results proves a conjecture of A. N. Kirillov about the uniqueness of the reduced form of a Coxeter type element in the bracket algebra of type C_n. We also study the bracket algebra of type D_n and show that a family of monomials has unique reduced forms in it. A special case of our results proves a conjecture of A. N. Kirillov about the uniqueness of the reduced form of a Coxeter type element in the bracket algebra of type D_n.
A Latin square $L(n,k)$ is a square of order $n$ with its entries colored with $k$ colors so that all the entries in a row or column have different colors. Let $d(L(n,k))$ be the minimal number of colored entries of an $n \times n$ square such that there is a unique way of coloring of the yet uncolored entries in order to obtain a Latin square $L(n, k)$. In this paper we discuss the properties of $d(L(n,k))$ for $k=2n-1$ and $k=2n-2$. We give an alternate proof of the identity $d(L(n, 2n-1))=n^2-n$, which holds for even $n$, and we establish the new result $d(L(n, 2n-2)) \geq n^2-\lfloor\frac{8n}{5}\rfloor$ and show that this bound is tight for $n$ divisible by 10.
A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v_1, ..., v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability, and motivated by the fact that k-orderedness of a graph implies (k-1)-connectivity, they posed the question of the existence of low degree k-ordered hamiltonian graphs. We construct an infinite family of graphs, which we call bracelet graphs, that are (k-1)-regular and are k-ordered hamiltonian for odd k. This result provides the best possible answer to the question of the existence of low degree k-ordered hamiltonian graphs for odd k. We further show that for even k, there exist no k-ordered bracelet graphs with minimum degree k-1 and maximum degree less than k+2, and we exhibit an infinite family of bracelet graphs with minimum degree k-1 and maximum degree k+2 that are k-ordered for even k. A concept related to k-orderedness, namely that of k-edge-orderedness, is likewise strongly related to connectivity properties. We study this relation in both undirected and directed graphs, and give bounds on the connectivity necessary to imply k-(edge-)orderedness properties.
A simple graph $G$ is \textit{k-ordered} (respectively, \textit{k-ordered hamiltonian}), if for any sequence of $k$ distinct vertices $v_1, ..., v_k$ of $G$ there exists a cycle (respectively, hamiltonian cycle) in $G$ containing these $k$ vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graphs other than $K_4$ and $K_{3, 3}$. Ng and Schultz observed that a 3-regular 4-ordered graph on more than 4 vertices is triangle free. We prove that a 3-regular 4-ordered graph $G$ on more than 6 vertices is square free, and we show that the smallest graph that is triangle and square free, namely the Petersen graph, is 4-ordered. Furthermore, we prove that the smallest graph after $K_4$ and $K_{3, 3}$ that is 3-regular 4-ordered hamiltonian is the Heawood graph, and we exhibit forbidden subgraphs for 3-regular 4-ordered hamiltonian graphs on more than 10 vertices. Finally, we construct an infinite family of 3-regular 4-ordered graphs.
A genus one labeled circle tree is a tree with its vertices on a circle, such that together they can be embedded in a surface of genus one, but not of genus zero. We define an e-reduction process whereby a special type of subtree, called an e-graph, is collapsed to an edge. We show that genus is invariant under e-reduction. Our main result is a classification of genus one labeled circle trees through e-reduction. Using this we prove a modified version of a conjecture of David Hough, namely, that the number of genus one labeled circle trees on $n$ vertices is divisible by $n$ or if it is not divisible by $n$ then it is divisible by $n/2$. Moreover, we explicitly characterize when each of these possibilities occur.