The Potts model representation and a Robinson–Schensted correspondence for the partition algebra

We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.     

On the stability of some integer programming algorithms

In the present paper we develop our approach for studying the stability of integer programming problems. We prove that the L-class enumeration method is stable on integer linear programming problems in the case of bounded relaxation sets [9]. The stability of some cutting plane algorithms is discussed.     

About the Notion of Semiring of Sets

We introduce a generalized notion of semiring and prove that all known properties that semirings have according to the old definition are preserved.     

Symmetry specified enumeration of substituted derivatives: an easy solution to the complex problem

Several sophisticated methods to solution of symmetry specified enumeration problems are available in the modern literature. In this paper we propose a simple technique that allows one to manually compute the exact numbers of fixed-symmetry derivatives for a given structure either with inclusion or ignoring the substitution patterns. The basic idea of the method suggested consists in the derivation of Pólya-like cycle indices for the automorphism groups of specially constructed orbit partition graphs; the expansion of these indices and subsequent simple calculations result in the desired numbers of substituted derivatives with achiral substituents. Limitations of the new technique (and a method suggested earlier) depend on the relevance of the orbit partitions for particular subgroups of the point symmetry group. For illustration purposes, the results obtained for the prismane (D _3h ) and adamantane (T _d ) structures are discussed. In the former case the numbers of substituted derivatives can be found for all subgroups of the D _3h group, whereas in the latter case these numbers can be determined for eight out of eleven subgroups of the T _d point symmetry group. This work is based on the text of the lecture presented by the authors at the 5th All-Russia Conference on Molecular Modeling (Moscow, April 2007). The paper deals with the methodology and detailed treatment of applied aspects related to solution of enumeration problems for substituted derivatives with prescribed symmetry groups. Unlike the known methods of symmetry specified enumeration, the technique suggested is simple enough and may be regarded as generalization of the Pólya methodology, which is widely used by chemists. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 227–245, February, 2008.     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

Enumeration of plane partitions and the algebraic Bethe anzatz

We establish a relation between an exactly solvable boson model and plane partitions, i.e., three-dimensional Young diagrams enclosed in a box of finite size, which allows representing the partition generating functions as correlation functions of an integrable model and deriving the MacMahon formulas for enumerating partitions using the quantum inverse scattering method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 193–203, February, 2007.     

Limit distribution of the coefficients of polynomials with only unit roots

We consider sequences of random variables whose probability generating functions have only roots on the unit circle, which has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth central and normalized (by the standard deviation) moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance is unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,707–738, 2015     

On Time Dynamics of Coagulation-Fragmentation Processes

We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time t≥0 are time homogeneous. Based on this, we provide a characterization of mean-field Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by-product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to mean-field models.     

A Polynomial Time Approximation Scheme for the Grade of Service Steiner Minimum Tree Problem

In this paper, we present the design of a Polynomial Time Approximation Scheme (PTAS) for the Grade of Service Steiner Minimum Tree (GOSST) problem, which is known to be NP-Complete. Previous research has focused on geometric analyses and different approximation algorithms have been designed. We propose a PTAS that provides a polynomial time, near-optimal solution with performance ratio 1+. The GOSST problem has some important applications. In network design, a fundamental issue for the physical construction of a network structure is the interconnection of many communication sites with the best choice of the connecting lines and the best allocation of the transmission capacities over these lines. Good solutions should provide paths with enough communication capacities between any two sites, with the least network construction costs. Also, the GOSST problem has applications in transportation, for road constructions and some potential uses in CAD in terms of interconnecting the elements on a plane to provide enough flux between any two elements.     

New weighted Rogers-Ramanujan partition theorems and their implications

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts (mod 6), and a set of three theorems involving partitions into parts (mod 7), .