• Corpus ID: 247084168

Refinements of the braid arrangement and two parameter Fuss-Catalan numbers

@inproceedings{Deshpande2022RefinementsOT,
  title={Refinements of the braid arrangement and two parameter Fuss-Catalan numbers},
  author={Priyavrat Deshpande and Krishna Menon and Writika Sarkar},
  year={2022}
}
A hyperplane arrangement inR is a finite collection of affine hyperplanes. Counting regions of hyperplane arrangements is an active research direction in enumerative combinatorics. In this paper, we consider the arrangement A n in R given by {xi = 0 | i ∈ [n]} ∪ {xi = a xj | k ∈ [−m,m], 1 ≤ i < j ≤ n} for some fixed a > 1. It turns out that this family of arrangements is closely related to the well-studied extended Catalan arrangement of type A. We prove that the number of regions of A n is a… 

Figures and Tables from this paper

References

SHOWING 1-10 OF 12 REFERENCES

Deformations of the braid arrangement and trees

Characteristic Polynomials of Subspace Arrangements and Finite Fields

Let A be any subspace arrangement in Rndefined over the integers and let Fqdenote the finite field withqelements. Letqbe a large prime. We prove that the characteristic polynomialχ(A, q) of A counts

Catalan Numbers

Sequences and arrays whose terms enumerate combinatorial structures have many applications in computer science. Knowledge (or estimation) of such integer-valued functions is, for example, needed in

Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley

A hyperplane arrangement is said to satisfy the “Riemann hypothesis” if all roots of its characteristic polynomial have the same real part. This property was conjectured by Postnikov and Stanley for

COMBINATORIAL ENUMERATION OF THE REGIONS OF SOME LINEAR ARRANGEMENTS

Richard Stanley suggested the problem of finding combinatorial proofs of formulas for counting regions of certain hyperplane arrangements defined by hyperplanes of the form xi = 0, xi = xj , and xi =

Functional composition patterns and power series reversion

found in the writings of Jacobson [9], Becker [2], Motzkin [11], and Bourbaki [3; 4]. This paper will be concerned with a natural generalization of Cayley's problem, and will show that the solution

On Noncrossing and Nonnesting Partitions for Classical Reflection Groups

TLDR
The number of noncrossing partitions of $\{1,2,\ldots,n\}$ with fixed block sizes has a simple closed form, given by Kreweras, and coincides with the corresponding number for nonnesting partitions, defined recently by Reiner and Postnikov.

Returns and Hills on Generalized Dyck Paths

TLDR
This paper finds that the probability that a randomly chosen ternary path has an even number of hills approaches 125/169 as the length of the path approaches infinity.

Concrete mathematics - a foundation for computer science

From the Publisher: This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid