A third type appears when one looks for a noncommutative generalization of applications related to the notion of characteristic polynomial of a matrix. Thus, according to whether one is interested in the construction of a polynomial with given roots or in the expansion of a product of linear factors, one has to consider two distinct specializations of the formal symmetric functions. ![]() For example, when a polynomial with noncommutative coefficients in one central variable is decomposed as a product of linear factors, the roots of these factors differ from those of the expanded polynomial. ![]() One arrives at different constructions, according to the particular kind of application under consideration. Next, we study the noncommutative analogs of symmetric polynomials. It also gives unified reinterpretation of a number of classical constructions. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing in descent algebras. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We consider partitioned graphs, by which we mean finite strongly connectedĭirected graphs with a partitioned edge set $ ^-)$-presentation, to which the We end by discussing some extensions of this work (limit laws, systems involving non-polynomial entire functions, algorithmic aspects). As a corollary, it shows that certain lattice paths and planar maps can not be generated by a context-free grammar (i.e., their generating function is not ℕ-algebraic). We extend what Philippe Flajolet called the Drmota-Lalley-Woods theorem (which is assuring α=-3/2 as soon as a “dependency graph” associated to the algebraic system defining the function is strongly connected): We fully characterize the possible critical exponents in the non-strongly connected case. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the appearing critical exponents α can not be 1/3 or -5/2, they in fact belong to a subset of dyadic numbers. Then, we study their asymptotics, known to be of the type f n ∼CA n n α. First, we recall the too-little-known fact that these coefficients f n have a closed form. ![]() This paper studies the coefficients of algebraic functions. We conjecture that the result holds for all sofic-Dyck shifts. is the generating series of some unambiguous context-free language. We prove that the zeta function of a finite-type-Dyck shift is a computable N-algebraic series, i.e. This proves that the zeta function of all sofic-Dyck shifts is a computable Z-algebraic series. We extend the formula to all sofic-Dyck shifts. An algebraic formula for the zeta function, which counts the periodic sequences of these shifts, can be obtained for sofic-Dyck shifts having a right-resolving presentation. A larger class of constraints, described by sofic-Dyck automata, are the visibly pushdown constraints whose corresponding set of biinfinite sequences are the sofic-Dyck shifts. Regular constraints are described by finite-state automata and the set of bi-infinite constrained sequences are finite-type or sofic shifts. ![]() sequences with a predefined set of properties. Constrained coding is a technique for converting unrestricted sequences of symbols into constrained sequences, i.e.
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