Related papers: Limit Densities of Patterns in Permutation Inflati…
The standard calculation of the spectrum of density perturbations produced during inflation assumes (i) |n-1| << 1 and (ii) |dn/dlnk| << |n-1|. Slow-roll predicts and observations require (i), but neither slow-roll nor observations require…
We prove that a nonsymmetric normal entry pattern of order $n$ ($n\ge 3$) has at most $n(n-3)/2+3$ distinct indeterminates and up to permutation similarity this number is attained by a unique pattern which is explicitly described.
Bounds on the entropy of patterns of sequences generated by independently identically distributed (i.i.d.) sources are derived. A pattern is a sequence of indices that contains all consecutive integer indices in increasing order of first…
A simple, discrete, parametric model is proposed to describe conditional (correlated) deposition of particles on a surface and formation of a connecting (percolating) cluster. The surface changes spontaneously its properties (phase…
This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of…
Let $K_n$ denote the number of distinct values among the first $n$ terms of an infinite exchangeable sequence of random variables $(X_1,X_2,\ldots)$. We prove for $n=3$ that the extreme points of the convex set of all possible laws of $K_3$…
The spectrum of primordial fluctuations from inflation can be obtained using a mathematically controlled, and systematically extendable, uniform approximation. Closed-form expressions for power spectra and spectral indices may be found…
Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this…
Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers…
In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as…
A combinatorial object is said to be quasirandom if it exhibits certain properties that are typically seen in a truly random object of the same kind. It is known that a permutation is quasirandom if and only if the pattern density of each…
We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set $A$ whose indicator function is 'approximately multiplicative' and uniformly distributed on short…
An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation…
Periodic point sets model all solid crystalline materials whose structures are determined in a rigid form and should be studied up to rigid motion or isometry preserving inter-point distances. In 2021 H.Edelsbrunner et al. introduced an…
Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute…
We look at geometric limits of large random non-uniform permutations. We mainly consider two theories for limits of permutations: permuton limits, introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion of scaling…
We derive the power spectrum $\mathcal P(k)$ of the density perturbations produced during inflation up to second-order corrections in the standard slow-roll approximation for an inflaton with more than one degree of freedom. We also present…
The basic workings of inflationary models are summarized, along with the arguments that strongly suggest that our universe is the product of inflation. I describe the quantum origin of density perturbations, giving a heuristic derivation of…
When two patterns occur equally often in a set of permutations, we say that these patterns are equipopular. Using both structural and analytic tools, we classify the equipopular patterns in the set of separable permutations. In particular,…
We examine the open problem of finding the shortest string that contains each of the n! permutations of n symbols as contiguous substrings (i.e., the shortest superpermutation on n symbols). It has been conjectured that the shortest…