Equations with a dyck language solution

In this paper we consider languages which are solutions of equations of the type X=XA1XB1X+⋅⋅⋅+XAnXBnX+Lλ. In cases where the solutions are unique these may be represented in terms of Dyck languages. We will also discuss equations with nonunique solutions

Parametrized Stochastic Grammars for RNA Secondary Structure Prediction

We propose a two-level stochastic context-free grammar (SCFG) architecture for parametrized stochastic modeling of a family of RNA sequences, including their secondary structure. A stochastic model of this type can be used for maximum a posteriori estimation of the secondary structure of any new sequence in the family. The proposed SCFG architecture models RNA subsequences comprising paired bases as stochastically weighted Dyck-language words, i.e., as weighted balanced-parenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phase-type distribution: that of the hitting time in a finite-state Markov chain. Without loss of generality, each such Markov chain can be taken to have a bounded complexity. The scheme yields an overall family SCFG with a manageable number of parameters.Comment: 5 pages, submitted to the 2007 Information Theory and Applications Workshop (ITA 2007

Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics

We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation with reflecting boundary conditions which is relevant to the Temperley--Lieb model of loops on a strip. By use of integral formulae we prove conjectures relating it to the weighted enumeration of Cyclically Symmetric Transpose Complement Plane Partitions and related combinatorial objects

Why Delannoy numbers?

This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems. These numbers appear in probabilistic game theory, alignments of DNA sequences, tiling problems, temporal representation models, analysis of algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of Statistical Planning and Inference

Density profiles in the raise and peel model with and without a wall. Physics and combinatorics

We consider the raise and peel model of a one-dimensional fluctuating interface in the presence of an attractive wall. The model can also describe a pair annihilation process in a disordered unquenched media with a source at one end of the system. For the stationary states, several density profiles are studied using Monte Carlo simulations. We point out a deep connection between some profiles seen in the presence of the wall and in its absence. Our results are discussed in the context of conformal invariance ($c = 0$ theory). We discover some unexpected values for the critical exponents, which were obtained using combinatorial methods. We have solved known (Pascal's hexagon) and new (split-hexagon) bilinear recurrence relations. The solutions of these equations are interesting on their own since they give information on certain classes of alternating sign matrices.Comment: 39 pages, 28 figure

Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte

Bounds on the norm of Wigner-type random matrices

We consider a Wigner-type ensemble, i.e. large hermitian $N\times N$ random matrices $H=H^*$ with centered independent entries and with a general matrix of variances $S_{xy}=\mathbb E|H_{xy}|^2$. The norm of $H$ is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of $S$ that substantially improves the earlier bound $2\| S\|^{1/2}_\infty$ given in [arXiv:1506.05098]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.Comment: 25 pages, 8 figure