Zeros of the Möbius function of permutations

We show that if a permutation \pi contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function \mu[1,\pi] of the interval [1,\pi] is zero. As a consequence, we prove that the proportion of permutations of length $\textit{n}$ with principal Möbius function equal to zero is asymptotically bounded below by (1\ -\ \sfrac{1}{e)^2} \geq 0.3995. This is the first result determining the value of \mu\left[1,\pi\right] for an asymptotically positive proportion of permutations \pi. We further establish other general conditions on a permutation \pi that ensure \mu\left[1,\pi\right]\ =\ 0, including the occurrence in \pi of any interval of the form \alpha\oplus\ 1\ \oplus\ \beta

Cyclic Permutations: Degrees and Combinatorial Types

This note will give elementary counts for the number of $n$-cycles in the permutation group ${\mathcal S}_n$ with a given degree (a variant of the descent number) and studies similar counting problems for the conjugacy classes of $n$-cycles under the action of the rotation subgroup of ${\mathcal S}_n$. This is achieved by relating such cycles to periodic orbits of an associated dynamical system acting on the circle. It is also shown that the distribution of degree on $n$-cycles is asymptotically normal as $n \to \infty$.Comment: 27 pages, 3 figures. New introduction and updated bibliograph

BOUNDING THE NUMBER OF COMPATIBLE SIMPLICES IN HIGHER DIMENSIONAL TOURNAMENTS

A tournament graph G is a vertex set V of size n, together with a directed edge set E ⊂ V × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, j ∈ V and (i, i) ∉ E for all i ∈ V. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the (n − 1)-dimensional simplex on V is given an orientation. In this dissertation we bound the number of compatible k-simplices, that is we bound the number of k-simplices such that its (k − 1)-faces with the already-specified orientation form an oriented boundary. We prove lower and upper bounds for all k ≥ 3. For k = 3 these bounds agree when the number of vertices n is q or q + 1 where q is a prime power congruent to 3 modulo 4. We also prove some lower bounds for values k \u3e 3 and analyze the asymptotic behavior

How round is a protein? Exploring protein structures for globularity using conformal mapping.

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry

PP-partitions and pp-positivity

Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse $P$-partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is $p$-positive whenever it is symmetric. As an application we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials and the more general $B$-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.Comment: 47 pages, 4 figure

Homomesies on permutations -- an analysis of maps and statistics in the FindStat database

In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer code rotation, the reverse, the complement, the Foata bijection, and the Kreweras complement. The statistics studied relate to familiar notions such as inversions, descents, and permutation patterns, and also more obscure constructs. Beside the many new homomesy results, we discuss our research method, in which we used SageMath to search the FindStat combinatorial statistics database to identify potential homomesies