Partition identities arising from theta function identities

The authors show that certain theta function identities of Schroeter and Ramanujan imply elegant partition identities.     

Frobenius Partitions in Extraspecial Groups

Let K be a field and G be the group of the upper unitriangular (n + 2) × ( n + 2) K-matrices with nonzero entries only in the first row and in the last column. Then G has a normal subgroup N with a complement H which are K-vector spaces respectively of dimensions n + 1 and n. In the present paper we show that the orbit of H under a group of automorphisms of G together with N, forms a partition of G, provided that there exists a commutative (possibly nonassociative) division algebra of dimension n + 1 over K. This algebra exists when K is a finite field.     

Elementary Proofs of Infinitely Many Congruences for 8-Cores

Using a very elementary argument, we prove the congruences where a₈(n) is the number of 8-core partitions of n. We also exhibit two infinite families of congruences modulo 2 for 8-cores.     

A General Algorithm for the MacMahon Omega Operator

In his famous book Combinatory Analysis MacMahon introduced Partition Analysis (Omega Calculus) as a computational method for solving problems in connection with linear diophantine inequalities and equations. The technique has recently been given a new life by G.E. Andrews and his coauthors, who had the idea of marrying it with the tools of to-days Computer Algebra.The theory consists of evaluating a certain type of rational function of the form A()^-1 B(1/)^-1 by the MacMahon operator. So far, the case where the two polynomials A and B are factorized as products of polynomials with two terms has been studied in details. In this paper we study the case of arbitrary polynomials A and B. We obtain an algorithm for evaluating the operator using the coefficients of those polynomials without knowing their roots. Since the program efficiency is a persisting problem in several-variable polynomial Calculus, we did our best to make the algorithm as fast as possible. As an application, we derive new combinatorial identities.AMS Subject Classification: 05A17, 05A19, 05E05, 15A15, 68W30.     

Extending Theorems of Göllnitz,A New Family of Partition Identities

A new family of partition identities is given which include as special cases two theorems of Göllnitz. We show, also, a relation between our result and a theorem given by Sylvester.     

Combinatoric enumeration of two-dimensional proper arrays

An n×mproper array is a two-dimensional rectangular array composed of directed cubes that obey certain constraints. Because of these constraints, the n×m proper arrays may be classified via a schema in which each n×m proper array is associated with a particular n×1 column. For a fixed n, the goal is to enumerate, modulo symmetry, all possible edge configurations associated with n×m proper arrays. By varying n, one constructs four combinatoric sequences, each of which enumerates a particular class of edge configurations. Convolution arguments and resultant calculations associate these sequences with cubic equations. These cubic equations allow one to predict M_n, the number of edge configurations, modulo symmetry, associated with n×m proper arrays.     

Analytical Distribution of Waiting Time in the M/{iD}/1 Queue

We give an analytical formula for the steady-state distribution of queue-wait in the M/G/1 queue, where the service time for each customer is a positive integer multiple of a constant D > 0. We call this an M/{iD}/1 queue. We give numerical algorithms to calculate the distribution. In addition, in the case that the service distribution is sparse, we give revised algorithms that can compute the distribution more quickly.AMS subject classification: 60K25, 90B22     

乘法分拆的计数函数的估值

设f(n)表示自然数n的乘法分拆数。对于所有奇数，较大地改进了n的系数，证明了：若n为奇数，则f(n)≤n/15 7/5。     

Phase coexistence in Ising, Potts and percolation models

We study phase coexistence (separation) phenomena in Ising, Potts and random cluster models in dimensions d3 below the critical temperature. The simultaneous occurrence of several phases is typical for systems with appropriately arranged (mixed) boundary conditions or for systems satisfying certain physically natural constraints (canonical ensembles). The various phases emerging in these models define a partition, called the empirical phase partition, of the space. Our main results are large deviations principles for (the shape of) the empirical phase partition. More specifically, we establish a general large deviation principle for the partition induced by large (macroscopic) clusters in the Fortuin–Kasteleyn model and transfer it to the Ising–Potts model where we obtain a large deviation principle for the empirical phase partition induced by the various phases. The rate function turns out to be the total surface free energy (associated with the surface tension of the model and with boundary conditions) which can be naturally assigned to each reasonable partition. These LDP-s imply a weak law of large numbers: asymptotically, the law of the phase partition is determined by an appropriate variational problem. More precisely, the empirical phase partition will be close to some partition which is compatible with the constraints imposed on the system and which minimizes the total surface free energy. A general compactness argument guarantees the existence of at least one such minimizing partition. Our results are valid for temperatures T below a limit of slab-thresholds conjectured to agree with the critical point T_c. Moreover, T should be such that there exists only one translation invariant infinite volume state in the corresponding Fortuin–Kasteleyn model; a property which can fail for at most countably many values and which is conjectured to be true for every T≠T_c.