Related papers: A Partition-Based Generating Function for Row-Conv…
The cogrowth series of a group with respect to a finite generating set is an important combinatorial quantity that seems very difficult to compute exactly, as evidenced by the scarcity of known examples. In this paper, we give a particular…
In this article we consider square permutations, a natural subclass of permutations defined in terms of geometric conditions, that can also be described in terms of pattern avoiding permutations, and convex permutoninoes, a related subclass…
We study a new class of polyominoes, called $p$-Fibonacci polyominoes, defined using $p$-Fibonacci words. We enumerate these polyominoes by applying generating functions to capture geometric parameters such as area, semi-perimeter, and the…
This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function,also…
This paper describes a novel method of generating hex-dominant meshes using pre-computed optimal subdivisions of the unit cube in a grid-based approach. Our method addresses geometries that are standard in mechanical engineering and often…
We prove that for any fixed d the generating function of the projection of the set of integer points in a rational d-dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice…
We introduce a framework for constructing fractal trees via analytic generator fields, replacing discrete affine transformations and symbolic rewriting rules by the integration of smooth vector fields in an internal state space. In this…
As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces:…
The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher…
In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that…
An integer partition \lambda of n corresponds, via its Ferrers diagram, to an artinian monomial ideal I of colength n in the polynomial ring on two variables. If the partition \lambda corresponds to an integrally closed ideal we call…
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two…
We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed Riemann-Hilbert approach to the computation of detailed asymptotics for these orthogonal…
We show that, for any fixed genus $g$, the ordinary generating function for the genus $g$ partitions of an $n$-element set into $k$ blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a…
A grid polygon is a polygon whose vertices are points of a grid. We define an injective map between permutations of length n and a subset of grid polygons on n vertices, which we call consecutive-minima polygons. By the kernel method, we…
We study the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank, and, using combinatorial and analytical techniques, show that this generating function is closely connected with…
Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting…
Let $\{a_\rr : \rr \in (\Z^+)^d \}$ be a $d$-dimensional array of numbers, for which the generating function $F(\zz) := \sum_\rr a_\rr \zz^\rr$ is meromorphic in a neighborhood of the origin. For example, $F$ may be a rational multivariate…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…
Ewens-Pitman's partition structure arises as a system of sampling consistent probability distributions on set partitions induced by the Pitman-Yor process. It is widely used in statistical applications, particularly in species sampling…