On Multiple Pattern Avoiding Set Partitions

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n, the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T. Our goal is to classify the equivalence classes among two-element pattern sets of several general types. First, we focus on pairs of patterns {\sigma,\tau}, where \sigma\ is a pattern of size three with at least two distinct symbols and \tau\ is an arbitrary pattern of size k that avoids \sigma. We show that pattern-pairs of this type determine a small number of equivalence classes; in particular, the classes have on average exponential size in k. We provide a (sub-exponential) upper bound for the number of equivalence classes, and provide an explicit formula for the generating function of all such avoidance classes, showing that in all cases this generating function is rational. Next, we study partitions avoiding a pair of patterns of the form {1212,\tau}, where \tau\ is an arbitrary pattern. Note that partitions avoiding 1212 are exactly the non-crossing partitions. We provide several general equivalence criteria for pattern pairs of this type, and show that these criteria account for all the equivalences observed when \tau\ has size at most six. In the last part of the paper, we perform a full classification of the equivalence classes of all the pairs {\sigma,\tau}, where \sigma\ and \tau\ have size four.Comment: 37 pages. Corrected a typ

Quenched Averages for self-avoiding walks and polygons on deterministic fractals

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W_n(S), and rooted self-avoiding polygons P_n(S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P_n(S), and W_n(S) for an arbitrary point S on the lattice. These are used to compute the averages $, ,$ and $<log W_n(S)>$ over different positions of S. We find that the connectivity constant $\mu$, and the radius of gyration exponent $\nu$ are the same for the annealed and quenched averages. However, $~ n log \mu + (\alpha_q -2) log n$, and $~ n log \mu + (\gamma_q -1)log n$, where the exponents $\alpha_q$ and $\gamma_q$ take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives $\alpha_q \simeq 0.72837 \pm 0.00001$; and $\gamma_q \simeq 1.37501 \pm 0.00003$, to be compared with the annealed values $\alpha_a = 0.73421$ and $\gamma_a = 1.37522$.Comment: 17 pages, 10 figures, submitted to Journal of Statistical Physic

Combinatorial generation via permutation languages. VI. Binary trees

In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung-Hoang-M\"utze-Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions

High-temperature expansion for Ising models on quasiperiodic tilings

We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyze periodic approximants by computing the partition function via the Kac-Ward determinant. For the critical properties, we find complete agreement with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two-dimensional lattices.Comment: 24 pages, 8 figures (EPS), uses IOP styles (included

Constraints on Primordial Magnetic Fields from Inflation

We present generic bounds on magnetic fields produced from cosmic inflation. By investigating field bounds on the vector potential, we constrain both the quantum mechanical production of magnetic fields and their classical growth in a model independent way. For classical growth, we show that only if the reheating temperature is as low as T_{reh} <~ 10^2 MeV can magnetic fields of 10^{-15} G be produced on Mpc scales in the present universe. For purely quantum mechanical scenarios, even stronger constraints are derived. Our bounds on classical and quantum mechanical scenarios apply to generic theories of inflationary magnetogenesis with a two-derivative time kinetic term for the vector potential. In both cases, the magnetic field strength is limited by the gravitational back-reaction of the electric fields that are produced simultaneously. As an example of quantum mechanical scenarios, we construct vector field theories whose time diffeomorphisms are spontaneously broken, and explore magnetic field generation in theories with a variable speed of light. Transitions of quantum vector field fluctuations into classical fluctuations are also analyzed in the examples.Comment: 26 pages, v2: published in JCA