An interesting combinatorial (enumeration) problem arises in the initial phase of the polyhedral homotopy continuation method for computing a11 so1utions of a polynomial equation system in complex variables. It is formulated as a problem of finding all solutions of a specially structured system of linear inequalities with a certain additional combinatorial condition. This paper presents a compu...

Finding the most stable tautomer or a set of low-energy tautomers of molecules is critical in many aspects of molecular modelling or virtual screening experiments. Enumeration of low-energy tautomers of neutral molecules in the gas-phase or typical solvents can be performed by applying available organic chemistry knowledge. This kind of enumeration is implemented in a number of software package...

We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a by-product of investigations on various combinatorial objects. For exa...

This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic expansion for the Taylor coefficients ofthe function. This approach is based on contour integration using Cauchy’s formula and Hankel-like contours. It constitutes an alternative to either Darboux’s method ...

We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrence...

Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are known to be related to the enumeration of chains in a new partial order on S∞, the universal k-Bruhat order. Here we present a monoid M for this order. We show that M is analogous to the nil-Coxeter monoid for the weak order on S∞. For this, we develop the theory of reduced sequences for M. We ...

We study a generalization of the Harish-Chandra–Itzykson–Zuber integral to tensors and its expansion in terms trace invariants two external tensors. This gives rise natural generalizations monotone double Hurwitz numbers, which count certain families constellations. find an expression these numbers simple thereby also providing expressions for arbitrary genus single ones. give interpretation di...

We introduce a unified framework to compute the solution space properties of broad class combinatorial optimization problems. These include finding one optimum solutions, counting number solutions given size, and enumeration sampling size. Using independent set problem as an example, we show how all these can be computed in approach generic tensor networks. demonstrate versatility this computat...