49 research outputs found
Combinatorics of the three-parameter PASEP partition function
We consider a partially asymmetric exclusion process (PASEP) on a finite
number of sites with open and directed boundary conditions. Its partition
function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to
be a generating function of permutation tableaux by the combinatorial
interpretation of Corteel and Williams.
We prove bijectively two new combinatorial interpretations. The first one is
in terms of weighted Motzkin paths called Laguerre histories and is obtained by
refining a bijection of Foata and Zeilberger. Secondly we show that this
partition function is the generating function of permutations with respect to
right-to-left minima, right-to-left maxima, ascents, and 31-2 patterns, by
refining a bijection of Francon and Viennot.
Then we give a new formula for the partition function which generalizes the
one of Blythe & al. It is proved in two combinatorial ways. The first proof is
an enumeration of lattice paths which are known to be a solution of the Matrix
Ansatz of Derrida & al. The second proof relies on a previous enumeration of
rook placements, which appear in the combinatorial interpretation of a related
normal ordering problem. We also obtain a closed formula for the moments of
Al-Salam-Chihara polynomials.Comment: 31 page
Refined enumeration of noncrossing chains and hook formulas
In the combinatorics of finite finite Coxeter groups, there is a simple
formula giving the number of maximal chains of noncrossing partitions. It is a
reinterpretation of a result by Deligne which is due to Chapoton, and the goal
of this article is to refine the formula. First, we prove a one-parameter
generalization, by the considering enumeration of noncrossing chains where we
put a weight on some relations. Second, we consider an equivalence relation on
noncrossing chains coming from the natural action of the group on set
partitions, and we show that each equivalence class has a simple generating
function. Using this we recover Postnikov's hook length formula in type A and
obtain a variant in type B.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1304.090
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report
Generalized Dumont-Foata polynomials and alternative tableaux
Dumont and Foata introduced in 1976 a three-variable symmetric refinement of
Genocchi numbers, which satisfies a simple recurrence relation. A six-variable
generalization with many similar properties was later considered by Dumont.
They generalize a lot of known integer sequences, and their ordinary generating
function can be expanded as a Jacobi continued fraction.
We give here a new combinatorial interpretation of the six-variable
polynomials in terms of the alternative tableaux introduced by Viennot. A
powerful tool to enumerate alternative tableaux is the so-called "matrix
Ansatz", and using this we show that our combinatorial interpretation naturally
leads to a new proof of the continued fraction expansion.Comment: 17 page
Bijections between pattern-avoiding fillings of Young diagrams
The pattern-avoiding fillings of Young diagrams we study arose from
Postnikov's work on positive Grassman cells. They are called Le-diagrams, and
are in bijection with decorated permutations. Other closely-related diagrams
are interpreted as acyclic orientations of some bipartite graphs. The
definition of the diagrams is the same but the avoided patterns are different.
We give here bijections proving that the number of pattern-avoiding filling of
a Young diagram is the same, for these two different sets of patterns. The
result was obtained by Postnikov via a reccurence relation. This relation was
extended by Spiridonov to obtain more general results about other patterns and
other polyominoes than Young diagrams, and we show that our bijections also
extend to more general polyominoes.Comment: 15 pages Version 2: important simplification and generalization of
the original bijection Version 3: small correction in references Version 4:
rewritten and submitte
Rook placements in Young diagrams and permutation enumeration
Given two operators and subject to the relation , and a word in and , the rewriting of in
normal form is combinatorially described by rook placements in a Young diagram.
We give enumerative results about these rook placements, particularly in the
case where . This case naturally arises in the context of the
PASEP, a random process whose partition function and stationary distribution
are expressed using two operators and subject to the relation
(matrix Ansatz). Using the link obtained by Corteel and Williams
between the PASEP, permutation tableaux and permutations, we prove a conjecture
of Corteel and Rubey about permutation enumeration. This result gives the
generating function for permutations of given size with respect to the number
of ascents and occurrences of the pattern 13-2, this is also the moments of the
-Laguerre orthogonal polynomials.Comment: V2. Many corrections. Submitte