A stochastic procedure for similarity searching in large virtual combinatorial libraries is presented. The method avoids explicit enumeration and calculation of descriptors for every virtual compound, yet provides an optimal or nearly optimal similarity selection in a reasonable time frame. It is based on the principle of probability sampling and the recognition that each reagent is represented...

We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P. The formula implies a simple combinatorial model for the characters of the irreducible representations of G and, more generally, for the Demazure characters. The construction is given in terms of a certain R-matrix, that is, a collection of operators satisfying the Yang-Baxte...

An overview of many of the major results in enumeration algorithms for Combinatorial Compositions can be found in this survey paper. The focus of this survey paper is on unrestricted compositions, bounded compositions, and future research in restricted compositions. The paper is concerned with generating these combinatorial objects, not counting the number of element under a composition. This p...

We consider the numbers arising in the problem of normal ordering of expressions in boson creation a† and annihilation a operators a ,a† =1 . We treat a general form of a boson string a† rnasn . . . a† r2as2 a† r1as1 which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions Dobiński-type formulas are obtained for thes...

Algebraic languages are at the heart of many successful optimization modeling systems, yet they have been used with only limited success for combinatorial (or discrete) optimization. We show in this paper, through a series of examples, how an algebraic modeling language might be extended to help with a greater variety of combinatorial optimization problems. We consider specifically those proble...

We consider the enumeration of partitions of a finite set according to the number of consecutive elements inside a block under the assumption that the elements are arranged around a circle. This statistic, commonly known as circular succession, continues to play a significant role in many combinatorial problems involving combinations of a set following its first appearance in a paper of Irving ...

The present article deals with differential equations with spectral parameter from the point of view of formal power series. The treatment does not make use of the notion of eigenvalue, but introduces a new idea: the spectral residue. The article focuses on second-order, self-adjoint problems. In such a setting every potential function determines a sequence of spectral residues. This correspond...

The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns. We give a new combinatorial interpretation for the Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the p-Catalan numbers in terms of p-ary t...

We address the six vertex model on a random lattice, which in combinatorial terms corresponds to enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved by Ivan Kostov 2000 using matrix integral techniques. convert Kostov's work argument involving functional equations coming from recursive decompositions maps, we...

The aim of the paper is to give a preliminary overview of some of the results of the thesis prepared by the author. We propose a bijective classification of the subgroups of the modular group by pointed trivalent diagrams. Conjugacy classes of those subgroups are in one to one correspondence with unpointed trivalent diagrams. We also give in the form of generating series, the number of those tr...