13,868 research outputs found
Set-polynomials and polynomial extension of the Hales-Jewett Theorem
An abstract, Hales-Jewett type extension of the polynomial van der Waerden
Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established:
Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring
of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a
\subset V and a nonempty set \gamma \subseteq {1,...,N} such that a \cap
(\gamma^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (\gamma^{d} x
{1}), a \cup (\gamma^{d} x {2}), ..., a \cup (\gamma^{d} x {q}) are all of the
same color.
This ``polynomial'' Hales-Jewett theorem contains refinements of many
combinatorial facts as special cases. The proof is achieved by introducing and
developing the apparatus of set-polynomials (polynomials whose coefficients are
finite sets) and applying the methods of topological dynamics.Comment: 43 pages, published versio
Double series representations for Schur's partition function and related identities
We prove new double summation hypergeometric -series representations for
several families of partitions, including those that appear in the famous
product identities of G\"ollnitz, Gordon, and Schur. We give several different
proofs for our results, using bijective partitions mappings and modular
diagrams, the theory of -difference equations and recurrences, and the
theories of summation and transformation for -series. We also consider a
general family of similar double series and highlight a number of other
interesting special cases.Comment: 19 page
A Reciprocity Theorem for Monomer-Dimer Coverings
The problem of counting monomer-dimer coverings of a lattice is a
longstanding problem in statistical mechanics. It has only been exactly solved
for the special case of dimer coverings in two dimensions. In earlier work,
Stanley proved a reciprocity principle governing the number of dimer
coverings of an by rectangular grid (also known as perfect matchings),
where is fixed and is allowed to vary. As reinterpreted by Propp,
Stanley's result concerns the unique way of extending to so
that the resulting bi-infinite sequence, for , satisfies a
linear recurrence relation with constant coefficients. In particular, Stanley
shows that is always an integer satisfying the relation where unless 2(mod 4) and
is odd, in which case . Furthermore, Propp's method is
applicable to higher-dimensional cases. This paper discusses similar
investigations of the numbers , of monomer-dimer coverings, or
equivalently (not necessarily perfect) matchings of an by rectangular
grid. We show that for each fixed there is a unique way of extending
to so that the resulting bi-infinite sequence, for , satisfies a linear recurrence relation with constant coefficients. We
show that , a priori a rational number, is always an integer, using a
generalization of the combinatorial model offered by Propp. Lastly, we give a
new statement of reciprocity in terms of multivariate generating functions from
which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete
Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes
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
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