dodgson condensation the historical and mathematical development of an experimental method abstract dodgson s condensation method has become a powerful tool in the automation of determinant evaluations in this expository paper i describe its 19 th century roots and the major steps on the path that began in the 20 th century when the iteration of an identity derived by dodgson first was studied including its role in the discovery of the alternating sign matrix conjecture the evaluation of an important 19 th century determinant in partition theory as well as a combinatorial proof of it i then discuss additional developments that have led the way to its use in modern experimental mathematics

more refined enumerations of alternating sign matrices abstract we study a further refinement of the standard refined enumeration of alternating sign matrices asms according to their first two rows instead of just the first row and more general d refined enumerations of asms according to the first d rows for the doubly refined case of d 2 we derive a system of linear equations satisfied by the doubly refined enumeration numbers a n i j that enumerate such matrices we give a conjectural explicit formula for a n i j and formulate several other conjectures about the sufficiency of the linear equations to determine the a n i j s and about an extension of the linear equations to the general d refined enumerations

the role of orthogonal polynomials in the six vertex model and its combinatorial applications the hankel determinant representations for the partition function and boundary correlation functions of the six vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory for specific values of the parameters of the model corresponding to 1 2 and 3 enumerations of alternating sign matrices asms these polynomials specialize to classical ones continuous hahn meixner pollaczek and continuous dual hahn respectively as a consequence a unified and simplified treatment of asm enumerations turns out to be possible leading also to some new results such as the refined 3 enumerations of asms furthermore the use of orthogonal polynomials allows us to express for generic values of the parameters of the model the partition function of the partially inhomogeneous model in terms of the one point boundary correlation functions of the homogeneous one

exact results for the critical behaviour of a nienhuis o n model on the square lattice exact results are obtained for a square lattice fifteen vertex model and related o n model recently proposed by nienhuis 1990 the partition function per site and finite size corrections are obtained along with the critical exponents via the temperley lieb equivalent six vertex model the exact exponents nu 4 7 and gamma 6 7 follow in the limit n 0 for even system sizes odd system sizes are also discussed the fifteen vertex model is also the vertex formulation of the anisotropic or q deformed spin 1 biquadratic model

generalized domino shuffling the problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting perfect matchings of some subgraph of an aztec diamond graph an or more generally calculating the sum of the weights of all the matchings where the weight of a matching is equal to the product of the pre assigned weights of the constituent edges assumed to be non negative this article presents efficient algorithms that work in this context to solve three problems finding the sum of the weights of the matchings of a weighted aztec diamond graph an computing the probability that a randomly chosen matching of an will include a particular edge where the probability of a matching is proportional to its weight and generating a matching of an at random the first of these algorithms is equivalent to a special case of mihai ciucu s cellular complementation algorithm j combin theory ser a 81 1998 34 and can be used to solve many of the same problems the second of the three algorithms is a generalization of not yet published work of alexandru ionescu and can be employed to prove an identity governing a three variable generating function whose coefficients are all the edge inclusion probabilities this formula has been used duke math j 85 1996 117 as the basis for asymptotic formulas for these probabilities but a proof of the generating function identity has not hitherto been published the third of the three algorithms is a generalization of the domino shuffling algorithm presented in j algebraic combin 1 1992 111 it enables one to generate random diabolo tilings of fortresses and thereby to make intriguing inferences about their asymptotic behavior

a class of interaction round a face models and its equivalence with an ice type model a new model called the temperley lieb interactions model is introduced in two dimensional lattice statistics on a square lattice the temperley lieb equivalence of this model to the six vertex self dual potts critical hard hexagons and critical nonintersecting string models is established a graphical equivalence of this model to the six vertex model generalizes this equivalence to noncritical cases of the above models the order parameters of a specialization of this model are studied

the arctic curve of the domain wall six vertex model the problem of the form of the arctic curve of the six vertex model with domain wall boundary conditions in its disordered regime is addressed it is well known that in the scaling limit the model exhibits phase separation with regions of order and disorder sharply separated by a smooth curve called the arctic curve to find this curve we study a multiple integral representation for the emptiness formation probability a correlation function devised to detect spatial transition from order to disorder we conjecture that the arctic curve for arbitrary choice of the vertex weights can be characterized by the condition of condensation of almost all roots of the corresponding saddle point equations at the same known value in explicit calculations we restrict to the disordered regime for which we have been able to compute the scaling limit of certain generating function entering the saddle point equations the arctic curve is obtained in parametric form and appears to be a non algebraic curve in general it turns into an algebraic one in the so called root of unity cases the arctic curve is also discussed in application to the limit shape of q enumerated with 0 q 4 large alternating sign matrices in particular as q 0 the limit shape tends to a nontrivial limiting curve given by a relatively simple equation

new results for the xx model with boundaries in this paper we study the xx model with particular non diagonal boundaries using a fermionization technique for an extended version of this chain we are able to present a new operator which commutes with the extended and also the original hamiltonian it is a highly non trivial conserved quantity in the spin chain formalism which gives rise to an interesting combinatorial partition function

a new spectrum extension method that maximizes the multistep minimum prediction error generalization of the maximum entropy concept given n 1 consecutive autocorrelations of a stationary discrete time stochastic process how this finite sequence is extended so that the power spectral density associated with the resulting infinite sequence of correlations is nonnegative everywhere is discussed it is well known that when the hermitian toeplitz matrix generated from the given autocorrelations is positive definite the problem has an infinite number of solutions and the particular solution that maximizes the entropy functional results in a stable all pole model of order n since maximization of the entropy functional is equivalent to maximization of the minimum mean square error associated with one step predictors the problem of obtaining admissible extensions that maximize the minimum mean square error associated with k step k

exact solution of the six vertex model with domain wall boundary conditions disordered phase the six vertex model or the square ice model with domain wall boundary conditions dwbc has been introduced and solved for finite n by korepin and izergin the solution is based on the yang baxter equations and it represents the free energy in terms of an n n hankel determinant paul zinn justin observed that the izergin korepin formula can be re expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction we use this observation to obtain the large n asymptotics of the six vertex model with dwbc in the disordered phase the solution is based on the riemann hilbert approach and the deift zhou nonlinear steepest descent method as was noticed by kuperberg the problem of enumeration of alternating sign matrices the asm problem is a special case of the six vertex model we compare the obtained exact solution of the six vertex model with known exact results for the 1 2 and 3 enumerations of asms and also with the exact solution on the so called free fermion line we prove the conjecture of zinn justin that the partition function of the six vertex model with dwbc has the asymptotics z n sim cn kappa e n 2 f as n and we find the exact value of the exponent

the quantum symmetric xxz chain at delta 1 2 alternating sign matrices and plane partitions we consider the ground state wavefunction of the quantum symmetric antiferromagnetic xxz chain with open and twisted boundary conditions at along with the ground state wavefunction of the corresponding o n loop model at n 1 based on exact results for finite size systems sums involving the wavefunction components and in some cases the largest component itself are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes

nonlinear wave equations as limits of convex minimization problems proof of a conjecture by de giorgi we prove a conjecture by de giorgi which states that global weak solutions of nonlinear wave equations such as w jwj p 2 w 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations these functionals which are integrals in space time of a convex lagrangian contain an exponential weight with a parameter and the initial data of the wave equation serve as boundary conditions as tends to zero the minimizers v converge up to subsequences to a solution of the nonlinear wave equation there is no restriction on the nonlinearity exponent and the method is easily extended to more general equations

on the partition function of the six vertex model with domain wall boundary conditions the six vertex model on an n n square lattice with domain wall boundary conditions is considered a fredholm determinant representation for the partition function of the model is given the kernel of the corresponding integral operator is of the so called integrable type and involves classical orthogonal polynomials from this representation a reconstruction formula is proposed which expresses the partition function as the trace of a suitably chosen quantum operator in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models

arctic circles domino tilings and square young tableaux the arctic circle theorem of jockusch propp and shor asserts that uniformly random domino tilings of an aztec diamond of high order are frozen with asymptotically high probability outside the arctic circle inscribed within the diamond a similar arctic circle phenomenon has been observed in the limiting behavior of random square young tableaux in this paper we show that random domino tilings of the aztec diamond are asymptotically related to random square young tableaux in a more refined sense that looks also at the behavior inside the arctic circle this is done by giving a new derivation of the limiting shape of the height function of a random domino tiling of the aztec diamond that uses the large deviation techniques developed for the square young tableaux problem in a previous paper by pittel and the author the solution of the variational problem that arises for domino tilings is almost identical to the solution for the case of square young tableaux by pittel and the author the analytic techniques used to solve the variational problem provide a systematic guess free approach for solving problems of this type which have appeared in a number of related combinatorial probability models

approximation algorithms for stochastic inventory control models we consider two classical stochastic inventory control models the periodic review stochastic inventory control problem and the stochastic lot sizing problem the goal is to coordinate a sequence of orders of a single commodity aiming to supply stochastic demands over a discrete finite horizon with minimum expected overall ordering holding and backlogging costs in this paper we address the important problem of finding computationally efficient and provably good inventory control policies for these models in the presence of correlated nonstationary time dependent and evolving stochastic demands this problem arises in many domains and has many practical applications in supply chain management our approach is based on a new marginal cost accounting scheme for stochastic inventory control models combined with novel cost balancing techniques specifically in each period we balance the expected cost of overordering i e costs incurred by excess inventory against the expected cost of underordering i e costs incurred by not satisfying demand on time this leads to what we believe to be the first computationally efficient policies with constant worst case performance guarantees for a general class of important stochastic inventory control models that is there exists a constant c such that for any instance of the problem the expected cost of the policy is at most c times the expected cost of an optimal policy in particular we provide a worst case guarantee of two for the periodic review stochastic inventory control problem and a worst case guarantee of three for the stochastic lot sizing problem our results are valid for all of the currently known approaches in the literature to model correlation and nonstationarity of demands over time

new enumeration formulas for alternating sign matrices and square ice partition functions abstract the refined enumeration of alternating sign matrices asms of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study starting with the refined alternating sign matrix conjecture of mills robbins rumsey 1983 25 its proof by zeilberger 1996 31 and more recent work on doubly refined and triply refined enumerations by several authors in this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the top left bottom triply refined and top left bottom right quadruply refined enumerations the latter case solves the problem of computing the full boundary correlation function for asms the enumeration formulas are proved by deriving new representations which are of independent interest for the partition function of the square ice model with domain wall boundary conditions at the combinatorial point 2 3

the importance of being odd in this letter i mainly consider a finite xxz spin chain with periodic boundary conditions and an odd number of sites this system is described by the hamiltonian hxxz j 1n jx j 1x jy j 1y jz j 1z as it turns out the ground state energy is proportional to the number of sites e 3n 2 for a special value of the asymmetry parameter 1 2 the trigonometric polynomial q u the zeros of which are parameters of the ground state bethe eigenvector is explicitly constructed this polynomial of degree n n 1 2 satisfies the baxter t q equation using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix it is possible to find a derivative of the ground state energy w r t the asymmetry parameter this derivative is closely connected with the correlation function jz j 1z 1 2 3 2n2 this correlation function is related to the average number of spin strings for the ground state nstring n 1 n i would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered

total positivity properties of generalized hypergeometric functions of matrix argument in multivariate statistical analysis several authors have studied the total positivity properties of the generalized 0f1 hypergeometric function of two real symmetric matrix arguments in this paper we make use of zonal polynomial expansions to obtain a new proof of a result that these 0f1 functions fail to satisfy certain pairwise total positivity properties this proof extends both to arbitrary generalized rfs functions of two matrix arguments and to the generalized hypergeometric functions of hermitian matrix arguments in the case of the generalized hypergeometric functions of two hermitian matrix arguments we prove that these functions satisfy certain modified pairwise tp2 properties the proofs of these results are based on sylvester s formula for compound determinants and the condensation formula of c l dodgson lewis carroll 1866

boundary correlation functions of the six vertex model we consider the six vertex model on an n n square lattice with the domain wall boundary conditions boundary one point correlation functions of the model are expressed as determinants of n n matrices generalizing the known result for the partition function in the free fermion case explicit answers are obtained the introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

the xxz spin chain at δ 1 2 bethe roots symmetric functions and determinants a number of conjectures have been given recently concerning the connection between the antiferromagnetic xxz spin chain at 1 2 and various symmetry classes of alternating sign matrices here we use the integrability of the xxz chain to gain further insight into these developments in doing so we obtain a number of new results using baxter s q function for the xxz chain for periodic twisted and open boundary conditions these include expressions for the elementary symmetric functions evaluated at the ground state solution of the bethe roots in this approach schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the bethe roots when evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices

manhattan lattice theta point exponents from kinetic growth walks and exact results from the nienhuis o n model kinetic growth walks kgw on the manhattan lattice have previously been shown to be equivalent to the static problem of interacting self avoiding walks on that lattice at the theta temperature here we illustrate how a complete set of exponents for the static problem including the crossover exponent phi and surface exponents at the ordinary and special transition points may be obtained from simulations of kinetic walks in the process we find that phi m 0 430 or 0 006 which encompasses the conjectured value 3 7 approximately 0 42857 for the theta point on an isotropic lattice our numerics confirm a predicted set of exponents for both the bulk and surface transitions in addition to results such as the exact internal energy at the bulk transition furthermore we point out that a recently examined variant of the nienhuis o n model can be mapped onto theta point walks on the manhattan lattice which allows identification of the scaling dimensions for that problem and thereby provides a method for proving all the numerical conjectures

proof of the macdonald conjecture a plane partition of n is an array a i not necessarily square of positive integers which has non increasing rows and columns such that the sum of its elements is n with such an array we associate the set d which is the set of all integer points i j k in the first octant such that aij is defined and 1 k aij a plane partition is cyclically symmetric if d is invariant under the mapping i j k o k i let m m n be the number of such cyclically symmetric partitions of n such that none of i j aij exceed m macdonald has conjectured that the generating function

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

matrix representation of copulas and quasi copulas defined on non square grids of the unit square in this paper a matrix representation for discrete copulas and quasi copulas defined on a non square grid i nxi m of 0 1 2 is given two special cases are studied irreducible discrete copulas and quasi copulas those that cannot be expressed as non trivial convex combination of other ones and those of minimal range this study is divided into two cases depending on whether n divides m or not in the first case it is proved that copulas and quasi copulas with minimal range admit a representation through a special kind of matrices that they are always irreducible and that this implication becomes an equivalence in the case of copulas moreover an algorithm to express any discrete copula as a convex combination of irreducible ones is given however the case when n does not divide m is quite more complex and it is proved that the previous results do not longer hold in this case similar results are given only for a subclass of copulas and quasi copulas those whose associated matrices are given by blocks

traveling wave solutions of diffusive lotka volterra equations a heteroclinic connection in we establish the existence of traveling wave solutions for a reaction dif fusion system based on the lotka volterra differential equation model of a predator and prey interaction the waves are of transition front type analogous to the solutions discussed by fisher and kolmogorov et al for a scalar reaction diffusion equation the waves discussed here are not necessarily monotone there is a speed c 0 such that for c c there is a traveling wave moving with speed c the proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a liapunov function to guarantee the existence of the traveling wave solution the traveling wave solution is equivalent to a heteroclinic orbit in 4 dimensional phase space

matrix representation of discrete quasi copulas in this paper discrete quasi copulas defined on a square grid i n 2 of 0 1 2 are studied and it is proved that they can be represented by means of a special class of matrices with entries in 1 1 special considerations are made for the case of irreducible discrete quasi copulas those with range i n showing that they can be represented through alternating sign matrices and that they generate all discrete quasi copulas through convex sums in the process the number of irreducible quasi copulas on i n is given and those functions d for which there exists a unique copula with d as its diagonal section are characterized

boundary correlation functions of the six and nineteen vertex models with domain wall boundary conditions abstract boundary correlation functions of the six and nineteen vertex models on an n n lattice with domain wall boundary conditions are studied the general expression of the boundary correlation functions is obtained for the six vertex model by the use of the quantum inverse scattering method for the nineteen vertex model the boundary correlation functions are shown to be expressed in terms of those for the six vertex model

connectivity patterns in loop percolation i the rationality phenomenon and constant term identities loop percolation also known as the dense o 1 loop model is a variant of critical bond percolation in the square lattice mathbb z 2 whose graph structure consists of a disjoint union of cycles we study its connectivity pattern which is a random noncrossing matching associated with a loop percolation configuration these connectivity patterns exhibit a striking rationality property whereby probabilities of naturally occurring events are dyadic rational numbers or rational functions of a size parameter n but the reasons for this are not completely understood we prove the rationality phenomenon in a few cases and prove an explicit formula expressing the probabilities in the cylindrical geometry as coefficients in certain multivariate polynomials this reduces the rationality problem in the general case to that of proving a family of conjectural constant term identities generalizing an identity due to di francesco and zinn justin our results make use of and extend algebraic techniques related to the quantum knizhnik zamolodchikov equation

boundary emptiness formation probabilities in the six vertex model at mathbf delta frac mathbf 1 mathbf 2 we define a new family of overlaps c n m for the xxz hamiltonian on a periodic chain of length n these are equal to the linear sums of the groundstate components in the canonical basis wherein m consecutive spins are fixed to the state we define the boundary emptiness formation probabilities as the ratios c n m c n 0 of these overlaps in the associated six vertex model they correspond to correlation functions on a semi infinite cylinder of perimeter n at the combinatorial point we obtain closed form expressions in terms of simple products of ratios of integers

spin chains and combinatorics in this paper we continue the investigation of finite xxz spin chains with periodic boundary conditions and odd number of sites initiated in our previous article stroganov yu g 2001 j phys a math gen 34 l179 85 as it turned out for a special value of the asymmetry parameter 1 2 the hamiltonian of the system has an eigenvalue which is exactly proportional to the number of sites e 3n 2 using mathematica we have found explicitly the corresponding eigenvectors for n 17 the obtained results support the conjecture our paper that this special eigenvalue corresponds to the ground state vector we make a lot of conjectures concerning the correlations of the model many remarkable relations between the wavefunction components are noted it turns out for example that the ratio of the largest component to the least one is equal to the number of the alternating sign matrices

on identities satisfied by minors of a matrix abstract the algebraic relations between minors of a generic matrix are of great interest in many domains invariant theory representations of the symmetric group algebraic geometry schur functions etc we show that most of these relations result easily from a particular identity which was stated by h w turnbull in 1909 but has remained little known since the introduction of cayley or peano algebras by g c rota throws light on this identity indeed turnbull s formula turns out to be nothing but the expression of associativity and anticommutativity for the second product the meet operation of these algebras a constant problem when dealing with minors identities is the choice of appropriate notations the difficulty of grasping the cumbersome symbols used by many authors of the last century is no doubt responsible for the present neglect of this field we propose to code these identities by permutations of letters in tableaux as an illustration of these principles we derive explicitly a great number of classical theorems due to sylvester bazin reiss picquet muir etc and we give also several new formulae which generalize some of these theorems however we do not try to give an exhaustive list of identities it seems to us more interesting to show that these tools provide an effective calculus on minors which enables anyone to find the formulae old or new suitable for a particular application

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

the octahedron recurrence and gln crystals abstract we study the hive model of g l n tensor products following knutson tao and woodward we define a coboundary category where the tensor product is given by hives and where the associator and commutor are defined using a modified octahedron recurrence we then prove that this category is equivalent to the category of crystals for the lie algebra g l n the proof of this equivalence uses a new connection between the octahedron recurrence and the jeu de taquin and schutzenberger involution procedures on young tableaux

schur polynomials and the yang baxter equation we describe a parametrized yang baxter equation with nonabelian parameter group that is we show that there is an injective map g mapsto r g from rm gl 2 mathbb c times rm gl 1 mathbb c to end v otimes v where v is a two dimensional vector space such that if g h in g then r 12 g r 13 gh r 23 h r 23 h r 13 gh r 12 g here r i j denotes r applied to the i j components of v otimes v otimes v the image of this map consists of matrices whose nonzero coefficients a 1 a 2 b 1 b 2 c 1 c 2 are the boltzmann weights for the non field free six vertex model constrained to satisfy a 1 a 2 b 1 b 2 c 1 c 2 0 this is the exact center of the disordered regime and is contained within the free fermionic eight vertex models of fan and wu as an application we show that with boundary conditions corresponding to integer partitions the six vertex model is exactly solvable and equal to a schur polynomial s times a deformation of the weyl denominator this generalizes and gives a new proof of results of tokuyama and hamel and king

applications of graphical condensation for enumerating matchings and tilings a technique called graphical condensation is used to prove various combinatorial identities among numbers of perfect matchings of planar bipartite graphs and tilings of regions graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph and then re partitioning the united matching actually a multigraph into matchings of two other subgraphs in one of two possible ways this technique can be used to enumerate perfect matchings of a wide variety of planar bipartite graphs applications include domino tilings of aztec diamonds and rectangles diabolo tilings of fortresses plane partitions and transpose complement plane partitions

boundary polarization in the six vertex model vertical arrow fluctuations near the boundaries in the six vertex model on the two dimensional n n square lattice with the domain wall boundary conditions are considered the one point correlation function boundary polarization is expressed via the partition function of the model on a sublattice the partition function is represented in terms of standard objects in the theory of orthogonal polynomials this representation is used to study the large n limit the presence of the boundary affects the macroscopic quantities of the model even in this limit the logarithmic terms obtained are compared with predictions from conformal field theory

the combinatorics of k marked durfee symbols andrews recently introduced k marked durfee symbols which are connected to moments of dyson s rank by these connections andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems the primary goal of this article is to provide combinatorial proofs in answer to andrews request we obtain a relation between k marked durfee symbols and durfee symbols by constructing bijections and all identities on k marked durfee symbols given by andrews could follow from this relation in a similar manner we also prove the identities due to andrews on k marked odd durfee symbols combinatorially which resemble ordinary k marked durfee symbols with a modified subscript and with odd numbers as entries

symmetry classes of alternating sign matrices under one roof in a previous article math co 9712207 we derived the alternating sign matrix asm theorem from the izergin korepin determinant for a partition function for square ice with domain wall boundary here we show that the same argument enumerates three other symmetry classes of alternating sign matrices vsasms vertically symmetric asms even htsasms half turn symmetric asms and even qtsasms quarter turn symmetric asms the vsasm enumeration was conjectured by mills the others by robbins math co 0008045 we introduce several new types of asms uasms asms with a u turn side uuasms two u turn sides osasms off diagonally symmetric asms oosasms off diagonally off antidiagonally symmetric and uosasms off diagonally symmetric with u turn sides uasms generalize vsasms while uuasms generalize vhsasms vertically and horizontally symmetric asms and another new class vhpasms vertically and horizontally perverse osasms oosasms and uosasms are related to the remaining symmetry classes of asms namely dsasms diagonally symmetric dasasms diagonally anti diagonally symmetric and tsasms totally symmetric asms we enumerate several of these new classes and we provide several 2 enumerations and 3 enumerations our main technical tool is a set of multi parameter determinant and pfaffian formulas generalizing the izergin korepin determinant for asms and the tsuchiya determinant for uasms solv int 9804010 we evaluate specializations of the determinants and pfaffians using the factor exhaustion method

approximation algorithm for the stochastic multiperiod inventory problem via a look ahead optimization approach we consider the single item multiperiod stochastic inventory problem with nonstationary and correlated demands we provide the first proof that a well known myopic policy which we call look ahead optimization la is in fact an approximation algorithm for solving the large dynamic program that arises from this problem we prove that la provides the tightest known approximation bound for this problem the expected cost of la is at most twice the expected holding cost plus the expected backorder cost of an optimal policy we introduce a new cost accounting technique and a new class of invariances on the ongoing performance of la relative to that of an optimal policy we use these invariances to extend the myopic optimality of la to a global bound on its performance we allow convex and nonlinear holding and backorder cost functions integral order quantities and positive lead times under a technical condition on the growth of demand we show in computational experiments that la has excellent average cas

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

provably near optimal balancing policies for multi echelon stochastic inventory control models we develop the first algorithmic approach to compute provably good ordering policies for a multi echelon stochastic inventory system facing correlated nonstationary and evolving demands over a finite horizon specifically we study the serial system our approach is computationally efficient and provides worst case guarantees that is the expected cost of the algorithms is guaranteed to be within a constant factor of the optimal expected cost depending on the assumption the constant varies between two and three our algorithmic approach is based on an innovative scheme to account for costs in a multi echelon multi period environment as well as repeatedly balancing between opposing cost the cost accounting scheme called a cause effect cost accounting scheme is significantly different from traditional cost accounting schemes in that it reallocates costs with the goal of assigning every unit of cost to the decision that caused the cost to be incurred we believe it will have additional applications

new results for directed vesicles and chains near an attractive wall in this paper we present new exact results for single fully directed walks and fully directed vesicles near an attractive wall this involves a novel method of solution for these types of problems the major advantage of this method is that it unlike many other single walker methods generalizes to an arbitrary number of walkers the method of solution involves solving a set of partial difference equations with a bethe ansatz the solution is expressed as a constant term formula which evaluates to sums of products of binomial coefficients the vesicle critical temperature is found at which a binding transition takes place and the asymptotic forms of the associated partition functions are found to have three different entropic exponents depending on whether the temperature is above below or at its critical value the expected number of monomers adsorbed onto the surface is found to become proportional to the vesicle length at temperatures below critical scaling functions near the critical point are determined

partition function of the eight vertex model with domain wall boundary condition we derive the recursive relations of the partition function for the eight vertex model on an n n square lattice with domain wall boundary condition solving the recursive relations we obtain the explicit expression of the domain wall partition function of the model in the trigonometric rational limit our results recover the corresponding ones for the six vertex model

arctic curves of the six vertex model on generic domains the tangent method we revisit the problem of determining the arctic curve in the six vertex model with domain wall boundary conditions we describe an alternative method by which we recover the previously conjectured analytic expression in the square domain we adapt the method to work for a large class of domains and for other models exhibiting limit shape phenomena we study in detail some examples and derive in particular the arctic curve of the six vertex model in a triangoloid domain at the ice point

a model of subdiffusive interface dynamics with a local conservation of minimum height we define a new model of interface roughening in one dimension which has the property that the minimum of interface height is conserved locally during the evolution this model corresponds to the limit q of the q color dimer deposition evaporation model introduced by us earlier hari menon and dhar j phys a math gen 28 6517 1995 we present numerical evidence from monte carlo simulations and the exact diagonalization of the evolution operator on finite rings that growth of correlations in this model is subdiffusive with dynamical exponent z 2 5 for periodic boundary conditions the variation of the gap in the relaxation spectrum with system size appears to involve a logarithmic correction term some generalizations of the model are briefly discussed

weighted energy dissipation functionals for doubly nonlinear evolution abstract this paper is concerned with the weighted energy dissipation wed functional approach to doubly nonlinear evolutionary problems this approach consists in minimizing wed functionals defined over entire trajectories we present the features of the wed variational formalism and analyze the related euler lagrange problems moreover we check that minimizers of the wed functionals converge to the corresponding limiting doubly nonlinear evolution finally we present a discussion on the functional convergence of sequences of wed functionals and present some application of the abstract theory to nonlinear pdes

chio s and dodgson s determinantal identities abstract in this expository paper we analyze and compare two determinantal identities constructed in the mid nineteenth century chio s and dodgson s from the perspective of their origins in earlier work by j j sylvester and c g j jacobi all the known proofs of chio s and dodgson s identities are cited and we present and compare them as modern algorithms for evaluating determinants restated as a recurrence we discuss the role of dodgson s identity in the development of the alternating sign matrix conjecture in the late twentieth century

determinant formula for the six vertex model with reflecting end using the quantum inverse scattering method for the xxz model with open boundary conditions we obtained the determinant formula for the six vertex model with reflecting end

domino tilings and the six vertex model at its free fermion point at the free fermion point the six vertex model with domain wall boundary conditions dwbc can be related to the aztec diamond a domino tiling problem we study the mapping on the level of complete statistics for general domains and boundary conditions this is obtained by associating with both models a set of non intersecting lines in the lindstrom gessel viennot lgv scheme one of the consequences for dwbc is that the boundaries of the ordered phases are described by the airy process in the thermodynamic limit

boundary k matrices for the six vertex and the n 2n 1 an 1 vertex models boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices k or theta for the six vertex model the general solution depending on four arbitrary parameters is found for the an 1 models all diagonal solutions are found the associated integrable magnetic hamiltonians are explicitly derived

nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators periodic problems where t is a given positive number a is the subdifferential of a timedependent lower semicontinuous convex function from h into 0 a with qf f co and where b t is a possibly nonmonotone multivalued nonlinear operator from d b t c h into h such that d c d b t m for all t e 0 t the case when b t is identically zero has been investigated by many authors see e g bknilan and b is 2 nagai 13 and yamada 20 as for the case when b t is not identically zero nor monotone however it seems that very little study has been devoted to this kind of problem on the other hand the cauchy problem for 1 1 has been studied by several authors i e attouch and damlamian 11 biroli 3 koi and watanabe 9 and the author 14 151 the main objective of the present paper is to pursue and develop the method employed in 151 for the periodic problem p p this paper is composed of five sections section 2 contains some preliminaries in section 3 we shall formulate main results in two distinctive types of theorems roughly speaking according to whether the external force f t can be taken arbitrarily large or not in section 4 we shall prove them by much the same method based on a schauder tychonoff type fixed point theorem as in 151 in section 5 it will be exemplified that our results give a 248 0022 0396184 3 00

a variational approach to symmetry monotonicity and comparison for doubly nonlinear equations abstract we advance a variational method to prove qualitative properties such as symmetries monotonicity upper and lower bounds sign properties and comparison principles for a large class of doubly nonlinear evolutionary problems including gradient flows some nonlocal problems and systems of nonlinear parabolic equations our method is based on the so called weighted energy dissipation wed variational approach this consists in defining a global parameter dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter goes to zero qualitative properties and comparison principles can be easily proved for minimizers of the wed functional and by passing to the limit for the limiting problem several applications of the abstract results to systems of nonlinear pdes and to fractional nonlocal problems are presented eventually we present some extensions of this approach in order to deal with rate independent systems and hyperbolic problems

a constant term approach to enumerating alternating sign trapezoids abstract we show that there is the same number of n l alternating sign trapezoids as there is of column strict shifted plane partitions of class l 1 with at most n parts in the top row thereby proving a result that was conjectured independently by behrend and aigner the first objects generalize alternating sign triangles which have recently been introduced by ayyer behrend and the author who showed that they are counted by the same formula as alternating sign matrices column strict shifted plane partitions of a fixed class were introduced in a slightly different form by andrews and they essentially generalize descending plane partitions they also correspond to cyclically symmetric lozenge tilings of a hexagon with a triangular hole in the center in addition we also provide three statistics on each class of objects and show that their joint distribution is the same we prove our result by employing a constant term approach that is based on the author s operator formula for monotone triangles this paper complements a forthcoming paper of behrend and the author where the six vertex model approach is used to show equinumeracy as well as generalizations involving statistics that are different from those considered in the present paper

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

alternating signed bipartite graphs and difference 1 colourings abstract we investigate a class of 2 edge coloured bipartite graphs known as alternating signed bipartite graphs asbgs that encode the information in alternating sign matrices the central question is when a given bipartite graph admits an asbg colouring a 2 edge colouring such that the resulting graph is an asbg we introduce the concept of a difference 1 colouring a relaxation of the concept of an asbg colouring and present a set of necessary and sufficient conditions for when a graph admits a difference 1 colouring the relationship between distinct difference 1 colourings of a particular graph is characterised and some classes of graphs for which all difference 1 colourings are asbg colourings are identified one key step is theorem 3 4 6 which generalises hall s matching theorem by describing a necessary and sufficient condition for the existence of a subgraph h of a bipartite graph in which each vertex v of h has some prescribed degree r v

profit estimation error in the newsvendor model under a parametric demand distribution we consider the newsvendor model in which uncertain demand is assumed to follow a probabilistic distribution with known functional form but unknown parameters these parameters are estimated unbia

a directed graph structure of alternating sign matrices abstract we introduce a new directed graph structure into the set of alternating sign matrices this includes bruhat graph bruhat order of the symmetric groups as a subgraph subposet drake gerrish skandera 2004 2006 6 7 gave characterizations of bruhat order in terms of total nonnegativity tnn and subtraction free laurent sfl expressions for permutation monomials with our directed graph we extend their idea in two ways first from permutations to alternating sign matrices second q analogs which we name q tnn and q sfl properties as a by product we obtain a new kind of permutation statistic the signed bigrassmannian statistics using dodgson s condensation on determinants

advanced determinant calculus a complement abstract this is a complement to my previous article advanced determinant calculus c krattenthaler advanced determinant calculus seminaire lotharingien combin 42 1999 the andrews festschrift article b42q 67 pp in the present article i share with the reader my experience of applying the methods described in the previous article in order to solve a particular problem from number theory g almkvist c krattenthaler j petersson some new formulas for experiment math 12 2003 441 456 moreover i add a list of determinant evaluations which i consider as interesting which have been found since the appearance of the previous article or which i failed to mention there including several conjectures and open problems

logarithmic torus amplitudes abstractfor the example of the logarithmic triplet theory at c 2 the chiral vacuumtorus amplitudes are analysed it is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations as well as a function thatcan be associated to the logarithmic extension of the vacuum representation a fewimplications and generalisations of this result are discussed 1 introduction during the last twenty years much has been understood about the struc ture of rational conformal eld theories rational conformal eld theories are characterisedby the property that they have only nitely many irreducible highest weight representationsof the chiral algebra or vertex operator algebra and that every highest weight represen tation is completely decomposable into irreducible representations the structure of thesetheories is well understood in particular the characters of the irreducible representationstransform into one another under modular transformations 1 see also 2 and the modulars matrix determines the fusion rules via the verlinde formula 3 a general proof for thishas only recently been given in 4 on the other hand it is clear that rational conformal eld theories are rather special andit is therefore important to understand the structure of more general classes of conformal eld theories one such class are the rational logarithmic theories that possess only nitelymany indecomposable representations but for which not all highest weight representationsare completely decomposable the name logarithmic comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts the rst example of a non rational logarithmic conformal eld theory was found in 5 see also 6 and the rstrational example that shall also concern us in this paper was constructed in 7 for somerecent reviews see 8 9 10 from a physics point of view logarithmic conformal eldtheories appear naturally in various models of statistical physics for example in the theory

refined cauchy littlewood identities and six vertex model partition functions iii deformed bosons abstract we study hall littlewood polynomials using an integrable lattice model of t deformed bosons working with row to row transfer matrices we review the construction of hall littlewood polynomials of the a n root system within the framework of this model introducing appropriate double row transfer matrices we extend this formalism to hall littlewood polynomials based on the b c n root system and obtain a new combinatorial formula for them we then apply our methods to prove a series of refined cauchy and littlewood identities involving hall littlewood polynomials the last two of these identities are new and relate infinite sums over hyperoctahedrally symmetric hall littlewood polynomials with partition functions of the six vertex model on finite domains

a variational principle for nonpotential perturbations of gradient flows of nonconvex energies abstract we investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in hilbert spaces we prove existence of solutions to elliptic in time regularizations of gradient flows by combining the minimization of a parameter dependent functional over entire trajectories and a fixed point argument these regularized solutions converge up to subsequences to solutions of the gradient flow as the regularization parameter goes to zero applications of the abstract theory to nonlinear reaction diffusion systems are presented

shared information in stationary states of stochastic processes we present four estimators of the shared information or interdepency in ground states given that the coefficients appearing in the wave function are all real non negative numbers and therefore can be interpreted as probabilities of configurations such ground states of hermitian and non hermitian hamiltonians can be given for example by superpositions of valence bond states which can describe equilibrium but also stationary states of stochastic models we consider in detail the last case the system being a classical not a quantum one using analytical and numerical methods we compare the values of the estimators in the directed polymer and the raise and peel models which have massive conformal invariant and nonconformal invariant massless phases we show that like in the case of the quantum problem the estimators verify the area law with logarithmic corrections when phase transitions take place

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

scaling fields in the two dimensional abelian sandpile model we consider the unoriented two dimensional abelian sandpile model from a perspective based on two dimensional conformal field theory we compute lattice correlation functions for various cluster variables at and off criticality from which we infer the field theoretic description in the scaling limit we find perfect agreement with the predictions of a c 2 conformal field theory and its massive perturbation thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory the question of the height 2 variable is also addressed with however no definite conclusion yet

a 2 approximation algorithm for stochastic inventory control models with lost sales in this paper we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst case performance guarantees in particular we introduce dual balancing policies for lost sales models that are conceptually similar to dual balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost that is in each period we balance two opposing costs the expected marginal holding costs against the expected marginal lost sales cost specifically we show that the dual balancing policies for the lost sales models provide a worst case performance guarantee of two under relatively general demand structures in particular the guarantee holds for independent not necessarily identically distributed demands and for models with correlated demands such as the ar 1 model and the multiplicative autoregressive demand model the policies and the worst case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times our analysis has several novel elements beyond the balancing ideas for backorder models

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

diagonally and antidiagonally symmetric alternating sign matrices of odd order we study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices dasasms of fixed odd order by introducing a case of the six vertex model whose configurations are in bijection with such matrices the model involves a grid graph on a triangle with bulk and boundary weights which satisfy the yang baxter and reflection equations we obtain a general expression for the partition function of this model as a sum of two determinantal terms and show that at a certain point each of these terms reduces to a schur function we are then able to prove a conjecture of robbins from the mid 1980 s that the total number of 2n 1 x 2n 1 dasasms is prod i 0 n 3i n i and a conjecture of stroganov from 2008 that the ratio between the numbers of 2n 1 x 2n 1 dasasms with central entry 1 and 1 is n n 1 among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980 s that for odd order dasasms is the last to have been proved

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

stochastic processes and conformal invariance we discuss a one dimensional model of a fluctuating interface with a dynamic exponent z 1 the events that occur are adsorption which is local and desorption which is nonlocal and may take place over regions of the order of the system size in the thermodynamic limit the time dependence of the system is given by characters of the c 0 logarithmic conformal field theory of percolation this implies in a rigorous way a connection between logarithmic conformal field theory and stochastic processes the finite size scaling behavior of the average height interface width and other observables are obtained the avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite size scaling with new critical exponents

determinants and alternating sign matrices let m be an n by n matrix by a connected minor of m of size k we mean a minor formed from k consecutive rows and k consecutive columns we give formulas for det m in terms of connected minors one involving minors of two consecutive sizes and one involving minors of three consecutive sizes the formulas express det m as sums indexed by sets of alternating sign matrices these matrices are described here and by w h mills d p robbins and h rumsey jr invent math 66 1982 73 87 j combin theory ser a 34 1983 340 359 the former study has led to the solution of macdonald s conjecture on cyclically symmetric plane partitions g e andrews invent math 53 1979 193 225 i g macdonald symmetric functions and hall polynomials p 53 oxford univ press clarendon oxford 1979

the pentagram map and y patterns abstract the pentagram map introduced by r schwartz is defined by the following construction given a polygon as input draw all of its shortest diagonals and output the smaller polygon which they cut out we employ the machinery of cluster algebras to obtain explicit formulas for the iterates of the pentagram map

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism

smooth phases roughening transitions and novel exponents in one dimensional growth models a class of solid on solid growth models with short range interactions and sequential updates is studied the models exhibit both smooth and rough phases in dimension d 1 some of the features of the roughening transition which takes place in these models are related to contact processes or directed percolation type problems the models are analyzed using a mean field approximation scaling arguments and numerical simulations in the smooth phase the symmetry of the underlying dynamics is spontaneously broken a family of order parameters which are not conserved by the dynamics is defined as well as conjugate fields which couple to these order parameters the corresponding critical behavior is studied and novel exponents identified and measured we also show how continuous symmetries can be broken in one dimension a field theory appropriate for studying the roughening transition is introduced and discussed

the raise and peel model of a fluctuating interface we propose a one dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter the adsorption rate at a special value of this parameter the model has some interesting features for example the spectrum is given by conformal field theory and the stationary non equilibrium probability distribution is given by the two dimensional equilibrium distribution of the ice model with domain wall type boundary conditions this connection is used to find exact analytic expressions for several quantities of the stochastic model vice versa some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model at the special point we study several properties of the model such as the height fluctuations as well as cluster and avalanche distributions the latter has a long tail which shows that the model exhibits self organized criticality we also find in the stationary state a special surface phase transition without enhancement and with a crossover exponent 2 3 furthermore we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken