In this paper, we present a general methodology to solve a wide variety of classical lattice path counting problems in a uniform way. These counting problems are related to Dyck paths, Motzkin paths and some generalizations. The methodology uses weighted automata, equations of ordinary generating functions and continued fractions. This new methodology is called Counting Automata Methodology. It...

This article describes recent developments connecting problems of enumerative combinatorics, constrained by linear systems of Diophantine inequalities, with number theory topics like partitions, partition congruences, and q-series identities. Special emphasis is put on the role of computer algebra algorithms. The presentation is intended for a broader audience; to this end, elementary introduct...

Normally a chess problem must have a unique solution, and is deemed unsound even if there are alternatives that differ only in the order in which the same moves are played. In an enumerative chess problem, the set of moves in the solution is (usually) unique but the order is not, and the task is to count the feasible permutations via an isomorphic problem in enumerative combinatorics. Almost al...

1. Introduction. The Lagrange inversion formula is one of the fundamental results of enumerative combinatorics. It expresses the coefficients of powers of the compositional inverse of a power series in terms of the coefficients of powers of the original power series. G. Labelle [10] extended Lagrange inversion to cycle index series, which are equivalent to symmetric functions. Although motivate...

If A is a finite set of cardinality n ≥ 1, 2 is the set of all subsets of A, and S is a nonempty subset of 2, we say that S has the odd-intersection property if there exists a subset N of A such that the cardinality of N ∩S is odd for each S ∈ S. Let OIP (n) denote the set of all subsets of 2 with the odd-intersection property. A nonempty set S of nonempty subsets of A is an obstruction (to the...

We study the bijective combinatorics of reduced words. These are fundamental objects in the study of Coxeter groups. We restrict our focus to reduced words of permutations and signed permutations. Our results can all be situated within the context of two parallels. The first parallel is between the enumerative theory of reduced words and that of Coxeter group elements. The second parallel is be...

We study a discrete attachment model for the self-assembly of polyhedra called the building game.We investigate two distinct aspects of the model: (i) enumerative combinatorics of the intermediate states and (ii) a notion of Brownian motion for the polyhedral linkage defined by each intermediate that we term conformational diffusion. The combinatorial configuration space of the model is compute...

The Schubert Calculus is a formal calculus of symbols representing geometric conditions used to solve problems in enumerative geometry. This originated in work of Chasles [9] on conics and was systematized and used to great effect by Schubert in his treatise “Kalkül der abzählenden Geometrie” [33]. The justification of Schubert’s enumerative calculus and the verification of the numbers he obtai...