33 research outputs found
Explicit monomial expansions of the generating series for connection coefficients
This paper is devoted to the explicit computation of generating series for
the connection coefficients of two commutative subalgebras of the group algebra
of the symmetric group, the class algebra and the double coset algebra. As
shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral
distribution of some random matrices that are of interest to statisticians.
Morales and Vassilieva (2009, 2011) found explicit formulas for these
generating series in terms of monomial symmetric functions by introducing a
bijection between partitioned hypermaps on (locally) orientable surfaces and
some decorated forests and trees. Thanks to purely algebraic means, we recover
the formula for the class algebra and provide a new simpler formula for the
double coset algebra. As a salient ingredient, we derive a new explicit
expression for zonal polynomials indexed by partitions of type [a,b,1^(n-a-b)]
Bijective enumeration of some colored permutations given by the product of two long cycles
Let be the permutation on symbols defined by $\gamma_n = (1\
2\...\ n)\betannp\gamma_n \beta^{-1}\frac{1}{n- p+1}\alpha\gamma_n\alphamn+1$, an
unexpected connection previously found by several authors by means of algebraic
methods. Moreover, our bijection allows us to refine the latter result with the
cycle type of the permutations.Comment: 22 pages. Version 1 is a short version of 12 pages, entitled "Linear
coefficients of Kerov's polynomials: bijective proof and refinement of
Zagier's result", published in DMTCS proceedings of FPSAC 2010, AN, 713-72
Bijective evaluation of the connection coefficients of the double coset algebra
This paper is devoted to the evaluation of the generating series of the
connection coefficients of the double cosets of the hyperoctahedral group.
Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a
partition , gives the spectral distribution of some random real matrices
that are of interest in random matrix theory. We provide an explicit evaluation
of this series when in terms of monomial symmetric functions. Our
development relies on an interpretation of the connection coefficients in terms
of locally orientable hypermaps and a new bijective construction between
locally orientable partitioned hypermaps and some permuted forests.Comment: 12 pages, 5 figure
Explicit generating series for connection coefficients
International audienceThis paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type .Cet article est dĂ©diĂ© au calcul explicite des sĂ©ries gĂ©nĂ©ratrices des constantes de structure de deux sous-algĂšbres commutatives de l'algĂšbre de groupe du groupe symĂ©trique, l'algĂšbre de classes et l'algĂšbre de double classe latĂ©rale. Tel que montrĂš par Hanlon, Stanley and Stembridge (1992), ces sĂ©ries dĂ©terminent la distribution spectrale de certaines matrices alĂ©atoires importantes en statistique. Morales et Vassilieva (2009, 2011) ont trouvĂš des formules explicites pour ces sĂ©ries gĂ©nĂ©ratrices en termes des monĂŽmes symĂ©triques en introduisant une bijection entre les hypercartes partitionnĂ©es sur des surfaces (localement) orientables et certains arbres et forĂȘts dĂ©corĂ©es. GrĂące Ă des moyens purement algĂ©briques, nous retrouvons la formule pour l'algĂšbre de classe et dĂ©terminons une nouvelle formule plus simple pour l'algĂšbre de double classe latĂ©rale. En tant que point saillant de notre dĂ©monstration nous calculons une formulation explicite pour les polynĂŽmes zonaux indexĂ©s par des partitions de type
The enriched -monomial basis of the quasisymmetric functions
We construct a new family of quasisymmetric functions for each element
of the base ring. We call them the "enriched -monomial quasisymmetric
functions". When is invertible, this family is a basis of
. It generalizes Hoffman's "essential quasi-symmetric
functions" (obtained for ) and Hsiao's "monomial peak functions" (obtained
for ), but also includes the monomial quasisymmetric functions as a
limiting case.
We describe these functions by several
formulas, and compute their products, coproducts and antipodes. The product
expansion is given by an exotic variant of the shuffle product which we call
the "stufufuffle product" due to its ability to pick several consecutive
entries from each composition. This "stufufuffle product" has previously
appeared in recent work by Bouillot, Novelli and Thibon, generalizing the
"block shuffle product" from the theory of multizeta values.Comment: 106 pages. A shortened version for more advanced readers will soon be
submitted. Comments are welcome
Performance analysis of modulation with diversity - A combinatorial approach II : Bijective methods
We propose different bijective methods for studying the performance of a modulation protocol with diversity in the context of mobile communications
The algebra of extended peaks
Building up on our previous works regarding -deformed -partitions, we
introduce a new family of subalgebras for the ring of quasisymmetric functions.
Each of these subalgebras admits as a basis a -analogue to Gessel's
fundamental quasisymmetric functions where is equal to a complex root of
unity. Interestingly, the basis elements are indexed by sets corresponding to
an intermediary statistic between peak and descent sets of permutations that we
call extended peak.Comment: 12 pages; extended abstract submitted for FPSAC. Longform papers on
the project are still forthcomin
Performance Evaluation of Demodulation Methods: a Combinatorial Approach
International audienceThis paper provides a combinatorial approach for analyzing the performance of demodulation methods used in GSM. We also show how to obtain combinatorially a nice specialization of an important performance evaluation formula, using its connection with a classical bijection of Knuth between pairs of Young tableaux and {0,1}-matrices