# Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

@article{Hackl2015AnalysisOB,
title={Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum},
author={Benjamin Hackl and Clemens Heuberger and H. Prodinger and Stephan G. Wagner},
journal={Annals of Combinatorics},
year={2015},
volume={20},
pages={775-797}
}
• Published 30 March 2015
• Mathematics
• Annals of Combinatorics
Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random…
7 Citations
The Bidirectional Ballot Polytope
• Mathematics
Integers
• 2018
It is proved that every $(2n-1)$-dimensional unit cube can be partitioned into isometric copies of the $\Theta(2^n/n)$, which forms a convex polytope sitting inside the unit cube, which is referred to as the bidirectional ballotpolytope.
On k-Dyck paths with a negative boundary
Paths that consist of up-steps of one unit and down-steps of $k$ units, being bounded below by a horizontal line $-t$, behave like $t+1$ ordered tuples of $k$-Dyck paths, provided that $t\le k$. We
Pattern avoiding permutations and involutions with a unique longest increasing subsequence
• Mathematics
• 2020
We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.
Some combinatorial matrices and their LU-decomposition
Abstract Three combinatorial matrices were considered and their LU-decompositions were found. This is typically done by (creative) guessing, and the proofs are more or less routine calculations.
Pattern Avoiding Permutations with a Unique Longest Increasing Subsequence
• Mathematics
Electron. J. Comb.
• 2020
An explicit formula for 231-avoiders is proved, it is shown that the growth rate for 321-avoiding permutations with a ULIS is 4, and it is proved that their generating function is not rational.
The LU-decomposition of Lehmer's tridiagonal matrix
The LU-decomposition of Lehmer's tridiagonal matrix is first guessed, then proved, which leads to an evaluation of the determinant.

## References

SHOWING 1-10 OF 19 REFERENCES
Identities for generalized Euler polynomials
• Mathematics
• 2014
For N∈ℕ, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers , defined as the coefficients in the expansion of 1/TN(1/z), are provided. These coefficients
The On-Line Encyclopedia of Integer Sequences
• N. Sloane
• Computer Science
Electron. J. Comb.
• 1994
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
Mellin Transforms and Asymptotics: Harmonic Sums
• Mathematics
Theor. Comput. Sci.
• 1995
A Course in Enumeration
Basics.- Fundamental Coefficients.- Formal Series and Infinite Matrices.- Methods.- Generating Functions.- Hypergeometric Summation.- Sieve Methods.- Enumeration of Patterns.- Topics.- The Catalan
On the Coefficients of the Asymptotic Expansion of n
Applying a theorem of Howard for a formula recently proved by Brassesco and M\'endez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion to the sequence of
NIST Handbook of Mathematical Functions
• Mathematics
• 2010
This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators and is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun.
Concrete mathematics - a foundation for computer science
• Education
• 1989
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