Related papers: Lattice Polynomials, 12312-Avoiding Partial Matchi…
The LLT polynomials $\mathcal{L}_{\mathbf{\beta}/\mathbf{\gamma}} (X;t)$ are a family of symmetric polynomials indexed by a tuple of (possibly skew-)partitions $\mathbf{\beta}/\mathbf{\gamma}=…
We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights…
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the…
We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include…
We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each vertex. For two Erd\H{o}s-R\'enyi graphs $\mathcal{G}(n,q)$ whose edges are…
In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove…
The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well…
We consider families $\mathcal{P}_n$ of plane lattice paths enumerated by Guy, Krattenthaler, and Sagan (1992). We show by explicit bijection that these families are equinumerous with the set $\mathrm{SYT}(n+2,2,1^n)$ of standard Young…
We introduce two classes of discrete polynomials and construct discrete equations admitting a Lax representation in terms of these polynomials. Also we give an approach which allows to construct lattice integrable hierarchies in its…
A matching of the set $[2n]=\{ 1,2,\ldots ,2n\}$ is a partition of $[2n]$ into blocks with two elements, i.e. a graph on $[2n]$ such that every vertex has degree one. Given two matchings $\sigma$ and $\tau$ , we say that $\sigma$ is a…
A lattice path inside the $m\times n$ table $T$ is a sequence $\nu_1,\ldots,\nu_k$ of cells such that $\nu_{j+1}-\nu_j\in\{(1,-1),(1,0),(1,1)\}$ for all $j=1,\ldots,k-1$. The number of lattice paths in $T$ from the first column to the…
We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain…
Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $\delta$ corresponds to the…
Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schroeder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight…
A gain graph is a graph whose edges are orientably labelled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For…
Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…
The notion of directed treewidth was introduced by Johnson, Robertson, Seymour and Thomas [Journal of Combinatorial Theory, Series B, Vol 82, 2001] as a first step towards an algorithmic metatheory for digraphs. They showed that some…
In this paper, we show that both 12312-avoiding partitions and 12321-avoiding partitions of the set $[n+1]$ are in one-to-one correspondence with Schr\"oder paths of semilength $n$ without peaks at even level. As a consequence, the refined…
We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary…
We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable…