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International audience; We introduce new $q$-tangent numbers based on the Carlitz $q$-analog of the Eulerian polynomial and the so-called doubloon combinatorial set-up. Those new $q$-tangent numbers are polynomials with positive integral coefficients. They are divisible by products of binomials of the form $1+q^i$, the quotients being $q$-analogs of the reduced tangent numbers having an explicit combinatorial interpretation.

International audience; The classical identity that relates Eulerian polynomials to tangent numbers together with the parallel result dealing with secant numbers is given a $q$-extension, both analytically and combinatorially. The analytic proof is based on a recent result by Shareshian and Wachs and the combinatorial one on the geometry of alternating permutations.

The flag-major index "fmaj" and the classical length function "$\ell$" are used to construct two $q$-analogs of the generating polynomial for the hyperoctahedral group~$B_n$ by number of positive and negative fixed points (resp. pixed points). Specializations of those $q$-analogs are also derived dealing with signed derangements and desarrangements, as well as several classical results that were previously proved for the symmetric group., Comment: 21 pages

La méthode du composé partitionnel, introduite en mathématiques combinatoires dans les années soixante-dix, a permis d'établir de nombreuses identités classiques sur les fonctions spéciales. On décrit ici cette méthode et l'on montre comment elle s'est écartée radicalement des méthodes utilisées jusque-là en analyse. The “composé partitionnel” method that was introduced in Combinatorial Mathematics in the seventies was used to derive several classical identities on Special Functions. Here we show how the method, which is further described below, has distanced itself altogether from the other methods traditionally used in analysis.

The statistics major index and inversion number, usually defined on ordinary words, have their counterparts in signed words, namely the so- called flag-major index and flag-inversion number. We give the construction of a new transformation on those signed words that maps the former statistic onto the latter one. It is proved that the transformation also preserves two other set-statistics: the inverse ligne of route and the lower records.

Two transformations are constructed that map the permutation group onto a well-defined subset of a partially commutative monoid generated by the so-called oiseaux. Those transformations are then used to show that some bivariable statistics introduced by Babson and Steingrímsson are Euler–Mahonian.

Le present article vise a proposer une description critique du corse parle d’aujourd’hui dans ses nouveaux domaines d’usage, notamment sur les ondes radiophoniques. Le corpus analyse concerne des journaux radiophoniques diffuses pendant environ une annee sur la radio locale, Radio Corse Frequenza Mora (frequence maure). La radio apparait comme le mini laboratoire d’un plurilinguisme original dont les journaux radiophoniques et leurs acteurs dessinent certaines orientations. Une attention particuliere a ete portee aux varietes de corse diffusees en les resituant dans leurs aires dialectales et en constatant leur dynamisme comme la complexite generee par le jeu des interactions langagieres.

Further computations are made on the traditional coupon collector’s problem when the collector shares his harvest with his younger brothers. When the book of the p-th brother of the collector is completed, the books of the younger brothers have certain numbers of empty spots. On the average, how many? Several answers can be brought to that question.

Babson and Steingrimsson have recently introduced seven new permutation statistics, that they conjectured were all Mahonian (i.e., equi-distributed with the number of inversions). We prove their conjecture for the first four and also prove that the first and the fourth are even Euler-Mahonian. We use two different, in fact, opposite, techniques. For three of them we give a computer-generated proof, using the Maple package ROTA, that implements the second author's ''Umbral Transfer Matrix Method.'' For the fourth one a geometric permutation transformation is used that leads to a further refinement of this Euler-Mahonian distribution study.

We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur’s identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.

The purpose of this paper is to give the description of a single algorithm that specializes into several classical transformations derived on words, namely the Cartier?Foata transform, its contextual extension due to Han and the twok-extensions proposed by Clarke and Foata.

International audience; The doubloon polynomials are generating functions for a class of combinatorial objects called normalized doubloons by the compressed major index. They provide a refinement of the $q$-tangent numbers and also involve two major specializations: the Poupard triangle and the Catalan triangle.

A true Tree Calculus is being developed to make a joint study of the two statistics “eoc” (end of minimal chain) and “pom” (parent of maximum leaf) on the set of secant trees. Their joint distribution restricted to the set {eoc-pom ≤ 1} is shown to satisfy two partial difference equation systems, to be symmetric and to be expressed in the form of an explicit three-variable generating function.

International audience; The finite difference equation system introduced by Christiane Poupard in the study of tangent trees is reinterpreted in the alternating permutation environment. It makes it possible to make a joint study of both tangent and secant trees and calculate the generating polynomial for alternating permutations by a new statistic, referred to as being the greater neighbour of the maximum.

Further transformations are derived on the symmetric group and related structures that extend previous works on the equidistributions of several classical univariable statistics.

International audience; The secant and tangent numbers are given (t, q)-analogs with an explicit combinatorial interpretation. This extends, both analytically and combinatorially, the classical evaluations of the Eulerian and Roselle polynomials at t = −1.

The polynomials commonly called “Eulerian” today have been introduced by Euler himself in his famous book “Institutiones calculi differentialis cum eius usu in analysi finitorum ac Doctrina serierum” [5, Chap. VII], back in 1755. They have been since thoroughly studied, extended, applied. The purpose of the present paper is to go back to Euler’s memoir, find out his motivation and reproduce his derivation, surprisingly partially forgotten. The rebirth of those polynomials in a q-environment is due to Carlitz two centuries after Euler. A brief overview of Carlitz’s method is given, as well as a short presentation of combinatorial works dealing with natural extensions of the classical Eulerian polynomials.

RésuméUn calcul explicite de distributions de statistiques d'ordre sur les permutations colorées est obtenu à l'aide de l'algèbre des fonctions de Schur.AbstractAn explicit calculation of distributions of order statistics on colored permutations is derived by means of the Schur function algebra.

We calculate the distribution of the sextuple statistic over the hyperoctahedral group Bn that involves the flag-excedance and flag-descent numbers “fexc” and “fdes,” the flag-major index “fmaj,” the positive and negative fixed point numbers “fix + ” and “fix ” and the negative letter number “neg.” Several specializations are considered. In particular, the joint distribution for the pair (fexc,fdes) is explicitly derived.

We construct a basis for the right quantum algebra introduced by Garoufalidis, Le and Zeilberger and give a method making it possible to go from an algebra submitted to commutation relations (without the variable q) to the right quantum algebra by means of an appropriate weight-function. As a consequence, a strong quantum MacMahon Master Theorem is derived. Besides, the algebra of biwords is systematically in use., Comment: 9 pages

Using classical transformations on the symmetric group and two transformations constructed in Fix-Mahonian Calculus I, we show that several multivariable statistics are equidistributed either with the triplet (fix,@?des,@?maj), or the pair (fix,@?maj), where ''fix,'' ''des'' and ''maj'' denote the number of fixed points, the number of descents and the major index, respectively.

We propose a new proof of the quantum version of MacMahon's Master Theorem, established by Garoufalidis, Le and Zeilberger., Comment: 8 pages

We construct two bijections of the symmetric group S"n onto itself that enable us to show that three new three-variable statistics are equidistributed with classical statistics involving the number of fixed points. The first one is equidistributed with the triplet (fix,des,maj), the last two with (fix,exc,maj), where ''fix'', ''des'', ''exc'' and ''maj'' denote the number of fixed points, the number of descents, the number of excedances and the major index, respectively.

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where "des" and "exc" stand for the number of descents and excedances, respectively., 26 pages

As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.

This paper takes up again the study of the Jacobi triple and Watson quintuple identities that have been derived combinatorially in several manners in the classical literature. It also contains a proof of the recent Farkas-Kra septuple product identity that makes use only of “smanipulatorics” methods.

RésuméOn the set of minimax trees of a given order there can be defined two families of operations, the complements and the reverses. We study the actions of those operations and show that their orbits are enumerated by combinatorial objects previously introduced, such as the Hetyei–Reiner trees, the increasing trees, and the André trees. Various generating functions for those trees by several statistics are also derived.

By practicing the philosophy of our beloved late master, Marco Schutzenberger, to whose memory this article is dedicated, we give an insightful bijective proof of the three-term recurrence satisfied by the Hipparchus-Schroeder numbers 1,1,3,11,45,197,903, ...

Capelli's and Turnbull's classical identities are given elegant combinatorial proofs., Plain TeX

The multilinear extensions of the Mehler formula found by Kibble, Slepian and Louck are shown to be equivalent. They can all be proved by using the combinatorial set-up of involutionary graphs, and so thus the classical Doetsch identity.

Nous decrivons trois modeles combinatoires pour les polynomes de Meixner m n ( x : β, c ), a savoir les endofonctions de Meixner, les permutations bicolorees et les permutations partiellement soulignees. Ce dernier modele a ete introduit par Kreweras, qui a etudie le cas particulier des polynomes m n ( x ; 1, 1/2).

The q q -tangent number T 2 n + 1 ( q ) {T_{2n + 1}}(q) is shown to be divisible by ( 1 + q ) a ( n , 1 ) ( 1 + q 2 ) a ( n , 2 ) ⋯ ( 1 + q n ) a ( n , n ) {(1 + q)^{a(n,1)}}{(1 + {q^2})^{a(n,2)}} \cdots {(1 + {q^n})^{a(n,n)}} , where the a ( n , i ) a(n,i) ’s are positive integers having the maximal property that a ( n , 1 ) + a ( n , 2 ) + ⋯ + a ( n , n ) = 2 n a(n,1) + a(n,2) + \cdots + a(n,n) = 2n whenever n n is a power of 2.

The q -secant number E 2 n ( q ) is shown to be congruent to q 2 n ( n -1) (its term of highest degree) mod( q + 1)2 by using the combinatorial set-up of alternating permutations.