arXiv subjects

Manda Riehl

Publications and source records attributed to Manda Riehl.

8 recordsLinked to original sources

arXiv2025

Accessibility of Modified NK Fitness Landscapes

In this paper we present two modifications of traditional $NK$ fitness landscapes, the $\theta NK$ and $HNK$ models, and explore these modifications via accessibility and ruggedness. The $\theta NK$ model introduces a parameter $\theta$ to integrate local Rough Mount Fuji-type correlations in subgenotype contributions, simulating more biologically realistic correlated fitness effects. The $HNK$ model incorporates gene regulation effects by introducing a masking mechanism where certain loci modulate the expression of other loci, simulating effects observed in gene regulatory networks without modeling the full network. Through extensive simulations across a wide range of parameters ($N$, $K$, $\theta$, and $H$), we analyze the impact of these modifications on landscape accessibility and the number of local optima. We find that increasing $\theta$ or the number of masking loci ($H$) generally enhances accessibility, even in landscapes with many local optima, showing that ruggedness doesn't necessarily hinder evolutionary pathways. Additionally, distinct interaction patterns (blocked, adjacent, random) lead to different observations in accessibility and optimum structure. While more complex than traditional $NK$, we believe each model provides a new biologically relevant facet to fitness landscapes and provides insight into how genetic and regulatory structures influence the evolutionary potential of populations.

q-bio.QM↗

arXiv2024

Predicting the stability of profiling signals of small RNAs

Profiling is a process that finds similarities between different RNA secondary structures by extracting signals from the Boltzmann sampling. The reproducibility of profiling can be identified by the standard deviation of number of features among Boltzmann samples. We found a strong relationship between the frequency of each helix class and its standard deviation of the frequency upon repeated Boltzmann sampling. We developed a perturbation technique to predict the stability of these featured helix classes without the need for repeated Boltzmann sampling, with accuracy between 84% and 94%, depending on the type of RNA. Our technique only requires 0.2% of the computation time compared to one profiling process.

q-bio.BM↗

arXiv2018

A bijection between the set of nesting-similarity classes and L & P matchings

Matchings are frequently used to model RNA secondary structures; however, not all matchings can be realized as RNA motifs. One class of matchings, called the L $\&$ P matchings, is the most restrictive model for RNA secondary structures in the Largest Hairpin Family (LHF). The L $\&$ P matchings were enumerated in $2015$ by Jefferson, and they are equinumerous with the set of nesting-similarity classes of matchings, enumerated by Klazar. We provide a bijection between these two sets. This bijection preserves noncrossing matchings, and preserves the sequence obtained reading left to right of whether an edge begins or ends at that vertex.

math.CO↗

arXiv2016

Pattern avoidance in forests of binary shrubs

We investigate pattern avoidance in permutations satisfying some additional restrictions. These are naturally considered in terms of avoiding patterns in linear extensions of certain forest-like partially ordered sets, which we call binary shrub forests. In this context, we enumerate forests avoiding patterns of length three. In four of the five non-equivalent cases, we present explicit enumerations by exhibiting bijections with certain lattice paths bounded above by the line $y=\ell x$, for some $\ell\in\mathbb{Q}^+$, one of these being the celebrated Duchon's club paths with $\ell=2/3$. In the remaining case, we use the machinery of analytic combinatorics to determine the minimal polynomial of its generating function, and deduce its growth rate.

math.CO↗

arXiv2016

Pattern Avoidance in Task-Precedence Posets

We have extended classical pattern avoidance to a new structure: multiple task-precedence posets whose Hasse diagrams have three levels, which we will call diamonds. The vertices of each diamond are assigned labels which are compatible with the poset. A corresponding permutation is formed by reading these labels by increasing levels, and then from left to right. We used Sage to form enumerative conjectures for the associated permutations avoiding collections of patterns of length three, which we then proved. We have discovered a bijection between diamonds avoiding 132 and certain generalized Dyck paths. We have also found the generating function for descents, and therefore the number of avoiders, in these permutations for the majority of collections of patterns of length three. An interesting application of this work (and the motivating example) can be found when task-precedence posets represent warehouse package fulfillment by robots, in which case avoidance of both 231 and 321 ensures we never stack two heavier packages on top of a lighter package.

math.CO↗

arXiv2016

Pattern Avoiding Linear Extensions of Rectangular Posets

Inspired by Yakoubov's 2015 investigation of pattern avoiding linear extensions of the posets called combs, we study pattern avoiding linear extensions of rectangular posets. These linear extensions are closely related to standard tableaux. For positive integers $s$ and $t$ we consider two natural rectangular partial orders on $\{1,2,\ldots,s t\}$, which we call the NE rectangular order and the EN rectangular order. First we enumerate linear extensions of both rectangular orders avoiding most sets of patterns of length three. Then we use both a generating tree and a bijection to show that the linear extensions of the EN rectangular order which avoid 1243 are counted by the Fuss-Catalan numbers. Next we use the transfer matrix method to enumerate linear extensions of the EN rectangular order which avoid 2143. Finally, we open an investigation of the distribution of the inversion number on pattern avoiding linear extensions.

math.CO↗

arXiv2009

On a pattern avoidance condition for the wreath product of cyclic groups with symmetric groups

In this paper, we extend to a non-consecutive case, the study of the pattern matching condition on the wreath product of the cyclic group and the symmetric group initiated by the authors in a previous paper. The main focus of our paper is (colored) patterns of length 2, although a number of enumerative results for longer patterns are also presented. A new non-trivial bijective interpretation for the Catalan numbers is found, in terms of simultaneously bi-avoiding two patterns in a wreath product.

math.CO↗

arXiv2009

New pattern matching conditions for wreath products of the cyclic groups with symmetric groups

We present several multi-variable generating functions for a new pattern matching condition on the wreath product of the cyclic group and the symmetric group. Our new pattern matching condition requires that the underlying permutations match in the usual sense of pattern matching for the symmetric group and that the corresponding sequence of signs match in the sense of words, rather than the exact equality of signs which has been previously studied. We produce the generating functions for the number of matches that occur in elements of the wreath product for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. We also provide multi-variable generating functions for the distribution of non-overlapping matches and for the number of elements of the wreath product which have exactly 2 matches which do not overlap for several patterns of length 2.

math.CO↗