Christophe Reutenauer

Counting permutations with given cycle structure and descent set

Abstract The number of permutations with given cycle structure and descent set is shown to be equal to the scalar product of two special characters of the symmetric group. Enumerative applications are given to cycles, involutions and derangements.

Recognizable formal power series on trees

Trees are a very basic object in computer science. They intervene in nearly a;ly domain, and they are studied for their own, or used to represent conveniently a given situation. There are at least three directions where investigations on trees themselves are motivated, and this for different reasons. First, the notion of tree is the basis of algebraic semantics (Nivat [19], Rosen [2*2], etc.). In this context, the study of special languages of trees (i.e. forests), their classification, and their behaviour under various types of transformations are of great importance (Arnold fl], Dauchet [8 J, L,ilin 1161, Mongy [18]). By essence, work in this area is an extension of the algebraic theory of languages; trees and ianguages are in fact directly related via the derivation trees of an algebraic grammar (Thatcher [25]). A second topic heavily related to trees concerns dzra structures, Trees, mainly binary trees &and its variants, constitute one of the most widely known data structures (see e.g. Hlnuth [15]). The analysis of the worst-caLe, expected or average running time t)ehzvGrr nfi’ certain algorithms requires sometimes long and delicate computations (Flajolet [lo], Kemp [14], Flajolet and Steyaert [12]). Finally, trees occupy a distinguished place in the enumeration of graphs and maps, both because of the simplicity of their structure and for the relationship between their el-lcodings and aigebraic languages. The nature of the enumerating series, and especially the question whether they are algebraic or not, is one of the central problems in this domain (Cori [7], Chottin [4]). We propose here a theory of formal power series on trees, and present some of their basic properties together with various examples of applications which, as we hope, will show the interest of its development within the framework we just sketched. A formal power series on trees is a function which Gssociates a number to esch tree. Thus we could also have called them ‘tree functions II in analogy with the term ‘word function’ used by several authors (Paz and Salomaa [ZO], Cobham [5]) as an equivalent denomination for formal power series on words. The main goal of a formal power series is to count, or to represent the result of some computation on

A decomposition of Solomon's descent algebra

Abstract A descent class, in the symmetric group Sn, is the collection of permutations with a given descent set. It was shown by L. Solomon (J. Algebra41 (1976), 255–264) that the product (in the group algebra Q(Sn)) of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q(Sn). We refer to it here as Solomons descent algebra and denote it by Σn. This algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices. The main goal of this paper is a decomposition of its multiplicative structure. It develops that Σn acts in a natural way on the so-called Lie monomials. This action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of Σn. In particular we obtain a natural basis of irreducible orthogonal idempotents Σλ (indexed by partitions of n) for the quotient Σ n √Σ n . Natural bases of nilpotents and idempotents for the subspaces EλΣnEμ, for two arbitrary partitions λ and μ, are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so-called decreasing factorization of an arbitary word into a product of Lyndon words.

Noncommutative rational series with applications

Preface Part I. Rational Series: 1. Rational series 2. Minimization 3. Series and languages 4. Rational expressions Part II. Arithmetic: 5. Automatic sequences and algebraic series 6. Rational series in one variable 7. Changing the semiring 8. Positive series in one variable Part III. Applications: 9. Matrix semigroups and applications 10. Noncommutative polynomials 11. Codes and formal series 12. Semisimple syntactic algebras Open problems and conjectures References Index of notation Index.

ON THE PALINDROMIC COMPLEXITY OF INFINITE WORDS

We study the problem of constructing infinite words having a prescribed finite set P of palindromes. We first establish that the language of all words with palindromic factors in P is rational. As a consequence we derive that there exists, with some additional mild condition, infinite words having P as palindromic factors. We prove that there exist periodic words having the maximum number of palindromes as in the case of Sturmian words, by providing a simple and easy to check condition. Asymmetric words, those that are not the product of two palindromes, play a fundamental role and an enumeration is provided.

Inversion of matrices over a commutative semiring

It is a well-known consequence of the elementary theory of vector spaces that if A and B are n-by-n matrices over a field (or even a skew field) such that AB = 1, then BA = 1. This result remains true for matrices over a commutative ring, however, it is not, in general, true for matrices over noncommutatives rings. In this paper we show that if A and B are n-by-n matrices over a commutative semiring, then the equation AB = 1 implies BA = 1. We give two proofs: one algebraic in nature, the other more combinatorial. Both proofs use a generalization of the familiar product law for determinants over a commutative semiring.

Lyndon words, free algebras and shuffles

1. Introduction. A Lyndon word is a primitive word which is minimum in its conjugation class, for the lexicographical ordering. These words have been introduced by Lyndon in order to find bases of the quotients of the lower central series of a free group or, equivalently, bases of the free Lie algebra [2], [7]. They have also many combinatorial properties, with applications to semigroups, pi-rings and pattern-matching, see [1], [10]. We study here the Poincaré-Birkhoff-Witt basis constructed on the Lyndon basis (PBWL basis). We give an algorithm to write each word in this basis: it reads the word from right to left, and the first encountered inversion is either bracketted, or straightened, and this process is iterated: the point is to show that each bracketting is a standard one: this we show by introducing a loop invariant (property (S)) of the algorithm. This algorithm has some analogy with the collecting process of P. Hall [5], but was never described for the Lyndon basis, as far we know. A striking consequence of this algorithm is that any word, when written in the PBWL basis, has coefficients in N (see Theorem 1). This will be proved twice in fact, and is similar to the same property for the Shirshov-Hall basis, as shown by M.P. Schutzenberger [11]. Our next result is a precise description of the dual basis of the PBWL basis. The former is denoted (S w), where w is any word, and we show that

The singular locus of a Schubert variety

Abstract The singular locus of a Schubert variety Xμ in the flag variety for GL n ( C ) is the union of Schubert varieties Xν, where ν runs over a set Sg(μ) of permutations in Sn. We describe completely the maximal elements of Sg(μ) under the Bruhat order, thus determining the irreducible components of the singular locus of Xμ.

On an involution of Christoffel words and Sturmian morphisms

There is a natural involution on Christoffel words, originally studied by the second author in [A. de Luca, Combinatorics of standard Sturmian words, Lecture Notes in Computer Science 1261 (1997) 249-267]. We show that it has several equivalent definitions: one of them uses the slope of the word, and changes the numerator and the denominator respectively in their inverses modulo the length; another one uses the cyclic graph allowing the construction of the word, by interpreting it in two ways (one as a permutation and its ascents and descents, coded by the two letters of the word, the other in the setting of the Fine and Wilf periodicity theorem); a third one uses central words and generation through iterated palindromic closure, by reversing the directive word. We show further that this involution extends to Sturmian morphisms, in the sense that it preserves conjugacy classes of these morphisms, which are in bijection with Christoffel words. The involution on morphisms is the restriction of some conjugation of the automorphisms of the free group. Finally, we show that, through the geometrical interpretation of substitutions of Arnoux and Ito, our involution is the same thing as duality of endomorphisms (modulo some conjugation).

Séries formelles et algèbres syntactiques

Abstract The notion of the syntactic monoid is well known to be very important for formal languages, and in particular for rational languages; examples of that importance are Kleenes theorem, Schutzenbergers theorem about aperiodic monoid and Eilenbergs theorem about varieties. We introduce here, for formal power series, a similar object: to each formal power series we associate its syntactic algebra. The Kleene-Schutzenberger theorem can then be stated in the following way: a series is rational if and only if its syntactic algebra has finite dimension. A rational central series (this means that the coefficient of a word depends only on its conjugacy class) is a linear combination of characters if and only if its syntactic algebra is semisimple. Fatou properties of rational series in one variable are extended to series in several variables and a special case of the rationality of the Hadamard quotient of two series is positively answered. The correspondence between pseudovarieties of finite monoids and varieties of rational languages, as studied by Eilenberg, is extended between pseudovarieties of finite dimensional algebras and varieties of rational series. We study different kinds of varieties that are defined by closure properties and prove a theorem similar to Schutzenbergers theorem on aperiodic monoids.