Some open problems on permutation patterns

This is a brief survey of some open problems on permutation patterns, with an emphasis on subjects not covered in the recent book by Kitaev, \emph{Patterns in Permutations and words}. I first survey recent developments on the enumeration and asymptotics of the pattern 1324, the last pattern of length 4 whose asymptotic growth is unknown, and related issues such as upper bounds for the number of avoiders of any pattern of length $k$ for any given $k$. Other subjects treated are the M\"obius function, topological properties and other algebraic aspects of the poset of permutations, ordered by containment, and also the study of growth rates of permutation classes, which are containment closed subsets of this poset.Comment: 20 pages. Related to upcoming talk at the British Combinatorial Conference 2013. To appear in London Mathematical Society Lecture Note Serie

The Möbius function of the permutation pattern Poset

A permutation \tau contains another permutation \sigma as a pattern if \tau has a subsequence whose elements are in the same order with respect to size as the elements in \sigma. This defines a partial order on the set of all permutations, and gives a graded poset P. We give a large class of pairs of permutations whose intervals in P have Mobius function 0. Also, we give a solution to the problem when \sigma occurs precisely once in \tau, and \sigma and \tau satisfy certain further conditions, in which case the Mobius function is shown to be either -1, 0 or 1. We conjecture that for intervals [\sigma,\tau] consisting of permutations avoiding the pattern 132, the magnitude of the Mobius function is bounded by the number of occurrences of \sigma in \tau. We also conjecture that the Mobius function of the interval [1,\tau] is -1, 0 or 1

Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).Comment: 17 pages, 10 figures; V2: minor changes (including title); V3: examples included (suggested by referee), to appear in "SIAM Journal on Discrete Mathematics

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Moebius function of an interval $[\sigma,\pi]$ in the poset of permutations ordered by pattern containment in the case where $\pi$ is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1, 2, ..., k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Moebius function in the case where $\sigma$ and $\pi$ are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142. A consequence of the formula is that the Moebius function of such an interval $[\sigma,\pi]$ is bounded by the number of occurrences of $\sigma$ as a pattern in $\pi$. We also show that for any separable permutation $\pi$ the Moebius function of $(1,\pi)$ is either 0, 1 or -1

Enumeration of polyominoes defined in terms of pattern avoidance or convexity constraints

In this thesis, we consider the problem of characterizing and enumerating sets of polyominoes described in terms of some constraints, defined either by convexity or by pattern containment. We are interested in a well known subclass of convex polyominoes, the k-convex polyominoes for which the enumeration according to the semi-perimeter is known only for k=1,2. We obtain, from a recursive decomposition, the generating function of the class of k-convex parallelogram polyominoes, which turns out to be rational. Noting that this generating function can be expressed in terms of the Fibonacci polynomials, we describe a bijection between the class of k-parallelogram polyominoes and the class of planted planar trees having height less than k+3. In the second part of the thesis we examine the notion of pattern avoidance, which has been extensively studied for permutations. We introduce the concept of pattern avoidance in the context of matrices, more precisely permutation matrices and polyomino matrices. We present definitions analogous to those given for permutations and in particular we define polyomino classes, i.e. sets downward closed with respect to the containment relation. So, the study of the old and new properties of the redefined sets of objects has not only become interesting, but it has also suggested the study of the associated poset. In both approaches our results can be used to treat open problems related to polyominoes as well as other combinatorial objects.Comment: PhD thesi

Prediction of Environmental Properties for Chlorophenols with Posetic Quantitative Super-Structure/Property Relationships (QSSPR)

Due to their widespread use in bactericides, insecticides, herbicides, andfungicides, chlorophenols represent an important source of soil contaminants. Theenvironmental fate of these chemicals depends on their physico-chemical properties. In theabsence of experimental values for these physico-chemical properties, one can use predictedvalues computed with quantitative structure-property relationships (QSPR). As analternative to correlations to molecular structure we have studied the super-structure of areaction network, thereby developing three new QSSPR models (poset-average, cluster-expansion, and splinoid poset) that can be applied to chemical compounds which can behierarchically ordered into a reaction network. In the present work we illustrate these posetQSSPR models for the correlation of the octanol/water partition coefficient (log Kow) and thesoil sorption coefficient (log KOC) of chlorophenols. Excellent results are obtained for allQSSPR poset models to yield: log Kow, r = 0.991, s = 0.107, with the cluster-expansionQSSPR; and log KOC, r = 0.938, s = 0.259, with the spline QSSPR. Thus, the poset QSSPRmodels predict environmentally important properties of chlorophenols

Recombination models forward and backward in time

Esser M. Recombination models forward and backward in time. Bielefeld: Universität Bielefeld; 2017

On The Möbius Function Of Permutations Under The Pattern Containment Order

We study several aspects of the Möbius function, μ[σ, π], on the poset of permutations under the pattern containment order. First, we consider cases where the lower bound of the poset is indecomposable. We show that μ[σ, π] can be computed by considering just the indecomposable permutations contained in the upper bound. We apply this to the case where the upper bound is an increasing oscillation, and give a method for computing the value of the Möbius function that only involves evaluating simple inequalities. We then consider conditions on an interval which guarantee that the value of the Möbius function is zero. In particular, we show that if a permutation π contains two intervals of length 2, which are not order-isomorphic to one another, then μ[1, π] = 0. This allows us to prove that the proportion of permutations of length n with principal Möbius function equal to zero is asymptotically bounded below by (1−1/e) 2 ≥ 0.3995. This is the first result determining the value of μ[1, π] for an asymptotically positive proportion of permutations π. Following this, we use “2413-balloon” permutations to show that the growth of the principal Möbius function on the permutation poset is exponential. This improves on previous work, which has shown that the growth is at least polynomial. We then generalise 2413-balloon permutations, and find a recursion for the value of the principal Möbius function of these generalisations. Finally, we look back at the results found, and discuss ways to relate the results from each chapter. We then consider further research avenues

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