1,255 research outputs found

### Enumeration of totally positive Grassmann cells

Alex Postnikov has given a combinatorially explicit cell decomposition of the
totally nonnegative part of a Grassmannian, denoted Gr_{kn}+, and showed that
this set of cells is isomorphic as a graded poset to many other interesting
graded posets. The main result of this paper is an explicit generating function
which enumerates the cells in Gr_{kn}+ according to their dimension. As a
corollary, we give a new proof that the Euler characteristic of Gr_{kn}+ is 1.
Additionally, we use our result to produce a new q-analog of the Eulerian
numbers, which interpolates between the Eulerian numbers, the Narayana numbers,
and the binomial coefficients.Comment: 21 pages, 10 figures. Final version, with references added and minor
corrections made, to appear in Advances in Mathematic

### Tableaux combinatorics for the asymmetric exclusion process

The partially asymmetric exclusion process (PASEP) is an important model from
statistical mechanics which describes a system of interacting particles hopping
left and right on a one-dimensional lattice of $n$ sites. It is partially
asymmetric in the sense that the probability of hopping left is $q$ times the
probability of hopping right. Additionally, particles may enter from the left
with probability $\alpha$ and exit from the right with probability $\beta$.
In this paper we prove a close connection between the PASEP and the
combinatorics of permutation tableaux. (These tableaux come indirectly from the
totally nonnegative part of the Grassmannian, via work of Postnikov, and were
studied in a paper of Steingrimsson and the second author.) Namely, we prove
that in the long time limit, the probability that the PASEP is in a particular
configuration $\tau$ is essentially the generating function for permutation
tableaux of shape $\lambda(\tau)$ enumerated according to three statistics. The
proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on
the {\it matrix ansatz} for the PASEP model.
As an application, we prove some monotonicity results for the PASEP. We also
derive some enumerative consequences for permutations enumerated according to
various statistics such as weak excedence set, descent set, crossings, and
occurences of generalized patterns.Comment: Clarified exposition, more general result, new author (SC), 19 pages,
6 figure

### The Matrix Ansatz, Orthogonal Polynomials, and Permutations

In this paper we outline a Matrix Ansatz approach to some problems of
combinatorial enumeration. The idea is that many interesting quantities can be
expressed in terms of products of matrices, where the matrices obey certain
relations. We illustrate this approach with applications to moments of
orthogonal polynomials, permutations, signed permutations, and tableaux.Comment: to appear in Advances in Applied Mathematics, special issue for
Dennis Stanto

### Combinatorial Hopf algebras, noncommutative Hall-Littlewood functions, and permutation tableaux

We introduce a new family of noncommutative analogues of the Hall-Littlewood
symmetric functions. Our construction relies upon Tevlin's bases and simple
q-deformations of the classical combinatorial Hopf algebras. We connect our new
Hall-Littlewood functions to permutation tableaux, and also give an exact
formula for the q-enumeration of permutation tableaux of a fixed shape. This
gives an explicit formula for: the steady state probability of each state in
the partially asymmetric exclusion process (PASEP); the polynomial enumerating
permutations with a fixed set of weak excedances according to crossings; the
polynomial enumerating permutations with a fixed set of descent bottoms
according to occurrences of the generalized pattern 2-31.Comment: 37 pages, 4 figures, new references adde

### Grass(mannian) trees and forests: Variations of the exponential formula, with applications to the momentum amplituhedron

© 2023 by the author(s). This work is made available under the terms of a Creative Commons Attribution License, available at https://creativecommons.org/licenses/by/4.0/The Exponential Formula allows one to enumerate any class of combinatorial objects built by choosing a set of connected components and placing a structure on each connected component which depends only on its size. There are multiple variants of this result, including Speicher’s result for noncrossing partitions, as well as analogues of the Exponential Formula for series-reduced planar trees and forests. In this paper we use these formulae to give generating functions for contracted Grassmannian trees and forests, certain graphs whose vertices are decorated with a helicity. Along the way we enumerate bipartite planar trees and forests, and we apply our results to enumerate various families of permutations: for example, bipartite planar trees are in bijection with separable permutations. It is postulated by Livia Ferro, Tomasz Łukowski and Robert Moerman (2020) that contracted Grassmannian forests are in bijection with boundary strata of the momentum amplituhedron, an object encoding the tree-level S-matrix of maximally supersymmetric Yang–Mills theory. With this assumption, our results give a rank generating function for the boundary strata of the momentum amplituhedron, and imply that the Euler characteristic of the momentum amplituhedron is 1.Peer reviewe

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