arXiv subjects

William Adamczak

Publications and source records attributed to William Adamczak.

4 recordsLinked to original sources

arXiv2025

Roller Coaster Permutations and Partition Numbers

This paper explores the partition properties of roller coaster permutations, a class of permutations characterized by maximizing the number of alternating runs in all subsequences. We establish a connection between the structure of these permutations and their partition numbers, defined as the minimum number of monotonic subsequences required to cover the permutation. Our main result provides a theoretical upper bound for the partition number of a roller coaster permutation of length $n$, given by $P_{max}(n) \le \lfloor\frac{\lceil\frac{n-2}{2}\rceil}{2}\rfloor + 2$. We further present experimental data for $n < 15$ that suggests this bound is nearly sharp.

math.CO↗

arXiv2016

A Note on the Structure of Roller Coaster Permutations

We consider the structure of roller coaster permutations as introduced by Ahmed & Snevily[1]. A roller coaster permutation is described as a permuta- tion that maximizes the total switches from ascending to descending or visa versa for the permutation and all of its subpermutations simultaneously. This paper looks at the alternating structure of these permutations and then we introduce a notion of a condition stronger than alternating for a permutation that we shall refer to as recursively alternating. We also examine the behav- ior of what entries can show up in even, odd, and end positions within the permutations.

math.CO↗

arXiv2007

Non-uniform mixing of quantum walk on cycles

A classical lazy random walk on cycles is known to mix to the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibit strong non-uniform mixing properties. Our results include the following: - The instantaneous distribution of a quantum walk on most even-length cycles is never uniform. - The average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform.

quant-ph↗

arXiv2003

A note on graphs resistant to quantum uniform mixing

Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the $n$-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs $K_{n}$, the continuous-time quantum walk is neither instantaneous (except for $n=2,3,4$) nor average uniform mixing (except for $n=2$). We explore two natural {\em group-theoretic} generalizations of the $n$-cube as a $G$-circulant and as a bunkbed $G \rtimes \Int_{2}$, where $G$ is a finite group. Analyses of these classes suggest that the $n$-cube might be special in having instantaneous uniform mixing and that non-uniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero.

quant-ph↗