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A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic ``strict fixed point, which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections.
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Branden and Claesson introduced mesh patterns to provide explicit expansions for certain permutation statistics as linear combinations of (classical) permutation patterns. The first systematic study of avoidance of mesh patterns was conducted by Hilmarsson et al., while the first systematic study of the distribution of mesh patterns was conducted by the first two authors. In this paper, we provide far-reaching generalizations for 8 known distribution results and 5 known avoidance results related to mesh patterns by giving distribution or avoidance formulas for certain infinite families of mesh patterns in terms of distribution or avoidance formulas for smaller patterns. Moreover, as a corollary to a general result, we find the distribution of one more mesh pattern of length 2.
A permutation $sigma$ describing the relative orders of the first $n$ iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is called an emph{order pattern}. For fixed $f$ and $n$, measuring the points $xin I$ (according to Lebesgue measure) that generate the order pattern $sigma$ gives a probability distribution $mu_n(f)$ on the set of length $n$ permutations. We study the distributions that arise this way for various classes of functions $f$. Our main results treat the class of measure preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each $n$ this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general $f$, apart from an obvious compatibility condition, there is no restriction on the sequence ${mu_n(f)}$ for $n=1,2,...$. In addition, we give a necessary condition for $f$ to have emph{finite exclusion type}, i.e., for there to be finitely many order patterns that generate all order patterns not realized by $f$. Using entropy we show that if $f$ is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then $f$ cannot have finite exclusion type. This generalizes results of S. Elizalde.
A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhangs four conjectures by constructing two involutions on permutations.
We investigate the distribution of bubble lifetimes and bubble lengths in DNA at physiological temperature, by performing extensive molecular dynamics simulations with the Peyrard-Bishop-Dauxois (PBD) model, as well as an extended version (ePBD) having a sequence-dependent stacking interaction, emphasizing the effect of the sequences guanine-cytosine (GC)/adenine-thymine (AT) content on these distributions. For both models we find that base pair-dependent (GC vs AT) thresholds for considering complementary nucleotides to be separated are able to reproduce the observed dependence of the melting temperature on the GC content of the DNA sequence. Using these thresholds for base pair openings, we obtain bubble lifetime distributions for bubbles of lengths up to ten base pairs as the GC content of the sequences is varied, which are accurately fitted with stretched exponential functions. We find that for both models the average bubble lifetime decreases with increasing either the bubble length or the GC content. In addition, the obtained bubble length distributions are also fitted by appropriate stretched exponential functions and our results show that short bubbles have similar likelihoods for any GC content, but longer ones are substantially more likely to occur in AT-rich sequences. We also show that the ePBD model permits more, longer-lived, bubbles than the PBD system.
Woodall proved that for a graph $G$ of order $ngeq 2k+3$ where $kgeq 0$ is an integer, if $e(G)geq binom{n-k-1}{2}+binom{k+2}{2}+1$ then $G$ contains a $C_{ell}$ for each $ellin [3,n-k]$. In this article, we prove a stability result of this theorem. As a byproduct, we give complete solutions to two problems in cite{GN19}. Our second part is devoted to an open problem by Nikiforov: what is the maximum $C$ such that for all positive $varepsilon<C$ and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius $rho(G)>sqrt{lfloorfrac{n^2}{4}rfloor}$ contains a cycle of length $ell$ for every $ellleq (C-varepsilon)n$. We prove that $Cgeqfrac{1}{4}$ by a method different from previous ones, improving the existing bounds. We also derive an ErdH{o}s-Gallai type edge number condition for even cycles, which may be of independent interest.
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