Related papers: Matrix Ansatz, lattice paths and rook placements
In this paper we consider an aggregation model f: X1 x ... x Xn --> Y for arbitrary sets X1, ..., Xn and a finite distributive lattice Y, factorizable as f(x1, ..., xn) = p(u1(x1), ..., un(xn)), where p is an n-variable lattice polynomial…
Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the…
Sums of walks for charged particles (e.g. Hofstadter electrons) on a square lattice in the presence of a magnetic field are evaluated. Returning loops are systematically added to directed paths to obtain the unrestricted propagators.…
Quenched reduction is revisited from the modern viewpoint of field-orbifolding. Fermions are included and it is shown how the old problem of preserving anomalies and field topology after reduction is solved with the help of the overlap…
In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More…
We show that the number of linear spaces on a set of $n$ points and the number of rank-3 matroids on a ground set of size $n$ are both of the form $(cn+o(n))^{n^2/6}$, where $c=e^{\sqrt 3/2-3}(1+\sqrt 3)/2$. This is the final piece of the…
We introduce a new approach to the enumeration of rational slope parking functions with respect to the area and a generalized dinv statistics, and relate the combinatorics of parking functions to that of affine permutations. We relate our…
We study the transition probabilities for the totally asymmetric simple exclusion process (TASEP) on the infinite integer lattice with a finite, but arbitrary number of first and second class particles. Using the Bethe ansatz we present an…
In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn…
The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A…
This paper examines the structure of poset matrices by formulating a set of new construction rules for this purpose. In this direction, the technique of partial composition operation will be introduced as the basis for the construction of…
We recall the main types of lattice paths, which are sequences in the lattice of integer coordinates points in the plane. We start with the fundamental central lattice paths and Dyck paths and proceed in elementary terms through recently…
We generalize the polynomial-time solvability of $k$-\textsc{Diverse Minimum s-t Cuts} (De Berg et al., ISAAC'23) to a wider class of combinatorial problems whose solution sets have a distributive lattice structure. We identify three…
Some of the classical orthogonal polynomials such as Hermite, Laguerre, Charlier, etc. have been shown to be the generating polynomials for certain combinatorial objects. These combinatorial interpretations are used to prove new identities…
We derive a path counting formula for two-dimensional lattice path model on a plane with filter restrictions. A filter is a line that restricts the path passing it to one of possible directions. Moreover, each path that touches this line is…
Proposed is an alternative method for permutational sampling in quantum gases using the path integral formulation of statistical mechanics. It is shown that in principle we are able to use two operators which enable us to construct a Markov…
Let $\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \in \{E,N\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \in…
We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for…
The celebrated Robinson-Schensted algorithm and each of its variants that have attracted substantial attention can be constructed using Fomin's "growth diagram" construction from a modular lattice that is also a weighted-differential poset.…
We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman--Stanley polytope of an n-tuple.