593 research outputs found
The Asymptotic Distribution of Symbols on Diagonals of Random Weighted Staircase Tableaux
Staircase tableaux are combinatorial objects that were first introduced due
to a connection with the asymmetric simple exclusion process (ASEP) and
Askey-Wilson polynomials. Since their introduction, staircase tableaux have
been the object of study in many recent papers. Relevant to this paper, the
distri- bution of parameters on the first diagonal was proven to be
asymptotically normal. In that same paper, a conjecture was made that the other
diagonals would be asymptotically Poisson. Since then, only the second and the
third diagonal were proven to follow the conjecture. This paper builds upon
those results to prove the conjecture for fixed k. In particular, we prove that
the distribution of the number of alphas (betas) on the kth diagonal, k > 1, is
asymptotically Poisson with parameter 1\2. In addition, we prove that symbols
on the kth diagonal are asymptotically independent and thus, collectively
follow the Poisson distribution with parameter 1
A simple bijection between permutation tableaux and permutations
We present a simple a bijection between permutations of with
descents and permutation tableaux of length with columns
An automaton-theoretic approach to the representation theory of quantum algebras
We develop a new approach to the representation theory of quantum algebras
supporting a torus action via methods from the theory of finite-state automata
and algebraic combinatorics. We show that for a fixed number , the
torus-invariant primitive ideals in quantum matrices can be seen as
a regular language in a natural way. Using this description and a semigroup
approach to the set of Cauchon diagrams, a combinatorial object that
paramaterizes the primes that are torus-invariant, we show that for fixed,
the number of torus-invariant primitive ideals in quantum matrices
satisfies a linear recurrence in over the rational numbers. In the case we give a concrete description of the torus-invariant primitive ideals
and use this description to give an explicit formula for the number P(3,n).Comment: 31 page
Overpartitions, lattice paths and Rogers-Ramanujan identities
We extend partition-theoretic work of Andrews, Bressoud, and Burge to
overpartitions, defining the notions of successive ranks, generalized Durfee
squares, and generalized lattice paths, and then relating these to
overpartitions defined by multiplicity conditions on the parts. This leads to
many new partition and overpartition identities, and provides a unification of
a number of well-known identities of the Rogers-Ramanujan type. Among these are
Gordon's generalization of the Rogers-Ramanujan identities, Andrews'
generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's
theorems for overpartitions.
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 sites. It is partially
asymmetric in the sense that the probability of hopping left is times the
probability of hopping right. Additionally, particles may enter from the left
with probability and exit from the right with probability .
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 is essentially the generating function for permutation
tableaux of shape 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
An iterative-bijective approach to generalizations of Schur's theorem
We start with a bijective proof of Schur's theorem due to Alladi and Gordon
and describe how a particular iteration of it leads to some very general
theorems on colored partitions. These theorems imply a number of important
results, including Schur's theorem, Bressoud's generalization of a theorem of
G\"ollnitz, two of Andrews' generalizations of Schur's theorem, and the
Andrews-Olsson identities.Comment: 16 page
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