For the most part, this article is a survey of concrete results in extremal combinatorics obtained with the method of flag algebras. But our survey is also preceded, interleaved and concluded with a few general digressions about the method itself. Also, instead of giving a plain and unannotated list of results, we try to divide our account into several connected stories that often include histo...

Combining a note by Rissanen and an idea of enumerative coding we obtain a new implementation of the Ziv-Lempel incremental parsing algorithm for coding and decoding discrete data sequences. Index Terms -Ziv-Lempel algorithm, enumerative coding.

Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen sub-model of N with |M | elements is isomorphic to M . Which asymptotic relations exist between t...

In this work we study numbers and polynomials generated by two type of composition of generating functions and get their explicit formulae. Furthermore we state an improvementof the composita formulae's given in [6] and [3], using the new composita formula's we construct a variety of combinatorics identities. This study go alone to dene new family of generalized Bernoulli polynomials which incl...

An enumerative invariant theory in Algebraic Geometry, Differential or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological $[E]=\alpha$ some geometric problem, using a virtual class $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ homology for moduli spaces ${\cal st}(\tau)\subseteq{\cal ss}(\tau)$ objects. Examples include Mochizu...