232 research outputs found
A Combinatorial Interpretation of the Free Fermion Condition of the Six-Vertex Model
The free fermion condition of the six-vertex model provides a 5 parameter
sub-manifold on which the Bethe Ansatz equations for the wavenumbers that enter
into the eigenfunctions of the transfer matrices of the model decouple, hence
allowing explicit solutions. Such conditions arose originally in early
field-theoretic S-matrix approaches. Here we provide a combinatorial
explanation for the condition in terms of a generalised Gessel-Viennot
involution. By doing so we extend the use of the Gessel-Viennot theorem,
originally devised for non-intersecting walks only, to a special weighted type
of \emph{intersecting} walk, and hence express the partition function of
such walks starting and finishing at fixed endpoints in terms of the single
walk partition functions
Three osculating walkers
We consider three directed walkers on the square lattice, which move
simultaneously at each tick of a clock and never cross. Their trajectories form
a non-crossing configuration of walks. This configuration is said to be
osculating if the walkers never share an edge, and vicious (or:
non-intersecting) if they never meet. We give a closed form expression for the
generating function of osculating configurations starting from prescribed
points. This generating function turns out to be algebraic. We also relate the
enumeration of osculating configurations with prescribed starting and ending
points to the (better understood) enumeration of non-intersecting
configurations. Our method is based on a step by step decomposition of
osculating configurations, and on the solution of the functional equation
provided by this decomposition
A refined Razumov-Stroganov conjecture
We extend the Razumov-Stroganov conjecture relating the groundstate of the
O(1) spin chain to alternating sign matrices, by relating the groundstate of
the monodromy matrix of the O(1) model to the so-called refined alternating
sign matrices, i.e. with prescribed configuration of their first row, as well
as to refined fully-packed loop configurations on a square grid, keeping track
both of the loop connectivity and of the configuration of their top row. We
also conjecture a direct relation between this groundstate and refined totally
symmetric self-complementary plane partitions, namely, in their formulation as
sets of non-intersecting lattice paths, with prescribed last steps of all
paths.Comment: 20 pages, 4 figures, uses epsf and harvmac macros a few typos
correcte
A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula
We provide yet another proof of the classical Lagrange-Good multivariable
inversion formula using techniques of quantum field theory.Comment: 9 pages, 3 diagram
Discrete integrable systems, positivity, and continued fraction rearrangements
In this review article, we present a unified approach to solving discrete,
integrable, possibly non-commutative, dynamical systems, including the - and
-systems based on . The initial data of the systems are seen as cluster
variables in a suitable cluster algebra, and may evolve by local mutations. We
show that the solutions are always expressed as Laurent polynomials of the
initial data with non-negative integer coefficients. This is done by
reformulating the mutations of initial data as local rearrangements of
continued fractions generating some particular solutions, that preserve
manifest positivity. We also show how these techniques apply as well to
non-commutative settings.Comment: 24 pages, 2 figure
On the functions counting walks with small steps in the quarter plane
Models of spatially homogeneous walks in the quarter plane
with steps taken from a subset of the set of jumps to the eight
nearest neighbors are considered. The generating function of the numbers of such walks starting at the origin and
ending at after steps is studied. For all
non-singular models of walks, the functions and are continued as multi-valued functions on having
infinitely many meromorphic branches, of which the set of poles is identified.
The nature of these functions is derived from this result: namely, for all the
51 walks which admit a certain infinite group of birational transformations of
, the interval of variation of splits into
two dense subsets such that the functions and are shown to be holonomic for any from the one of them and
non-holonomic for any from the other. This entails the non-holonomy of
, and therefore proves a conjecture of
Bousquet-M\'elou and Mishna.Comment: 40 pages, 17 figure
Summation and transformation formulas for elliptic hypergeometric series
Using matrix inversion and determinant evaluation techniques we prove several
summation and transformation formulas for terminating, balanced,
very-well-poised, elliptic hypergeometric series.Comment: 21 pages, AMS-LaTe
Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams
We consider the problem of finding the number of matrices over a finite field
with a certain rank and with support that avoids a subset of the entries. These
matrices are a q-analogue of permutations with restricted positions (i.e., rook
placements). For general sets of entries these numbers of matrices are not
polynomials in q (Stembridge 98); however, when the set of entries is a Young
diagram, the numbers, up to a power of q-1, are polynomials with nonnegative
coefficients (Haglund 98).
In this paper, we give a number of conditions under which these numbers are
polynomials in q, or even polynomials with nonnegative integer coefficients. We
extend Haglund's result to complements of skew Young diagrams, and we apply
this result to the case when the set of entries is the Rothe diagram of a
permutation. In particular, we give a necessary and sufficient condition on the
permutation for its Rothe diagram to be the complement of a skew Young diagram
up to rearrangement of rows and columns. We end by giving conjectures
connecting invertible matrices whose support avoids a Rothe diagram and
Poincar\'e polynomials of the strong Bruhat order.Comment: 24 pages, 9 figures, 1 tabl
Multivariate Lagrange inversion formula and the cycle lemma
International audienceWe give a multitype extension of the cycle lemma of (Dvoretzky and Motzkin 1947). This allows us to obtain a combinatorial proof of the multivariate Lagrange inversion formula that generalizes the celebrated proof of (Raney 1963) in the univariate case, and its extension in (Chottin 1981) to the two variable case. Until now, only the alternative approach of (Joyal 1981) and (Labelle 1981) via labelled arborescences and endofunctions had been successfully extended to the multivariate case in (Gessel 1983), (Goulden and Kulkarni 1996), (Bousquet et al. 2003), and the extension of the cycle lemma to more than 2 variables was elusive. The cycle lemma has found a lot of applications in combinatorics, so we expect our multivariate extension to be quite fruitful: as a first application we mention economical linear time exact random sampling for multispecies trees
Enumeration of simple random walks and tridiagonal matrices
We present some old and new results in the enumeration of random walks in one
dimension, mostly developed in works of enumerative combinatorics. The relation
between the trace of the -th power of a tridiagonal matrix and the
enumeration of weighted paths of steps allows an easier combinatorial
enumeration of the paths. It also seems promising for the theory of tridiagonal
random matrices .Comment: several ref.and comments added, misprints correcte
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