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 3-Factorizations of an N-Cycle

This paper is dedicated to the factorizations of the symmetric group. Introducing a new bijection for partitioned 3-cacti, we derive an el- egant formula for the number of factorizations of a long cycle into a product of three permutations. As the most salient aspect, our construction provides the first purely combinatorial computation of this number

Bijective enumeration of some colored permutations given by the product of two long cycles

Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations$\beta$of$n$in which the numbers from 1 to$n$are colored with$p$colors such that two elements in a same cycle have the same color. We show that the proportion of colored permutations such that$\gamma_n \beta^{-1}$is a long cycle is given by the very simple ratio$\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects such as partitioned hypermaps and thorn trees. This formula is actually equivalent to the proportionality of the number of long cycles$\alpha$such that$\gamma_n\alpha$has$m$cycles and Stirling numbers of size$n+1$, 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 $\nu$, 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 $\nu=(n)$ 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

A syntactic analysis of nominal and pronominal associative plurals

An associative plural is a nominal expression that refers to a group by naming its most salient member (1). The construction is used to introduce a new group into discourse, a group that is understood to be inherently (or contextually) associated with its named protagonist. (1) Pa-hulle (Afrikaans, den Besten 1996:16) Dad-them ‘Dad and Mum\u27 or \u27Dad and his folks’ In this paper, I argue for an analysis of associative plurals as phrasal expressions where the protagonist and the group are two separate syntactic entities. Namely, I suggest that associatives are headed by a non-descriptive nominal with group semantics. The reference of this group is determined through its association with the protagonist. The protagonist is a referential modifier which starts out in a modifier projection and moves to the specifier of DP. I begin by showing that associative protagonists share a number of syntactic and morphological properties with other types of referential modifiers such as demonstratives, personal pronouns and certain types of possessives. I go on to demonstrate that languages employ different strategies in spelling out the functional features of the non-descriptive group nominal, and that the apparent surface diversity of associative marking can be derived from the same syntactic structure. Finally, I suggest that my analysis of associatives can be extended to personal pronouns in their associative, anaphoric, and non-canonical interpretations

Feminism and Eternal Feminine: The Case of a Happy Union

The dozen or so women in this [feminist] group called Woman and Russia, produced one issue of the journal of the same title when a split between them occurred resulting in two groups, Woman and Russia and Maria. The former was headed by Tatyana Mamonova, writer and the most active contributor to the first issue, an enthusiastic supporter of many ideas of the contemporary Western feminism. Expelled from the Soviet Union for their activities in 1980, the authors of both journals continued working in the West. Subsequently Mamonova was warmly welcomed by the Western feminist Organizations, while Maria took a stance unpalatable both to the Soviet authorities and to most of the Western feminism. Its authors (Tatyana Goricheva, Yulia Voznesenskaya, Natal'ya Malakhovskaya, Galina Grigorieva, and others) preached a return to the traditional family ideals which were to find their justification in the religious and spiritual tradition of the Russian people. The freedoms and rights granted by the Soviet legislation were announced to have resulted in a near-catastrophic state of moral deterioration, to which Western feminism could give no solutions; instead the "new Russian feminism" was to be pursued. The "new Russian feminism" had painfully little new to offer to Russian women - its ideals were obedience to God, spiritual responsibility for the family, and fulfillment in motherhood. Soon Maria members disposed of the "feminist" allegiance of their group entirely and preferred to associate themselves with the politically neutral "women's cause" or "women's solidarity". Why did this commitment attract a prevailing majority within the group as well as the sympathies of fellow dissidents, whereas Mamonova was supported by very few in her country? This work will try to elucidate the content of the "new Russian feminism", i.e. the religious feminism of the Maria group, as well as the reasons for its popularity with the Russian intellectuals. The events of the end of 1979 - beginning of 1980, i.e. the publication of both groups' journals, the intervention of the KGB and the subsequent expulsion of the major contributors from the Soviet Union on the eve of the Olympic games, were extensively covered in the West in 1980-1981. However, the entire undertaking was dismissed as a courageous but very short-lived and theoretically obscure attempt at articulating female grievances against the Soviet system and its patriarchal character. While indeed detached from feminist theoretical discussions and largely rooted in experience, Woman and Russia was described by one Western publication as the "howl" of Russian women. It is one of the objectives of the present work to demonstrate the roots of the feminist publications, and, especially those of Maria, in the intellectual and political concerns of the post-war generations of Russian intellectuals

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 $[a,b,1^{n-a-b}]$.Cet article est dédié au calcul explicite des séries génératrices des constantes de structure de deux sous-algèbres commutatives de l'algèbre de groupe du groupe symétrique, l'algèbre de classes et l'algèbre de double classe latérale. Tel que montrè par Hanlon, Stanley and Stembridge (1992), ces séries déterminent la distribution spectrale de certaines matrices aléatoires importantes en statistique. Morales et Vassilieva (2009, 2011) ont trouvè des formules explicites pour ces séries génératrices en termes des monômes symétriques en introduisant une bijection entre les hypercartes partitionnées sur des surfaces (localement) orientables et certains arbres et forêts décorées. Grâce à des moyens purement algébriques, nous retrouvons la formule pour l'algèbre de classe et déterminons une nouvelle formule plus simple pour l'algèbre de double classe latérale. En tant que point saillant de notre démonstration nous calculons une formulation explicite pour les polynômes zonaux indexés par des partitions de type $[a,b,1^{n-a-b}]$