Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

Introduction to $q$-analogues and symmetric functions Macdonald polynomials and the space of diagonal harmonics The $q, t$-Catalan numbers The $q, t$-Schroder polynomial Parking functions and the Hilbert series The shuffle conjecture The proof of the $q, t$-Schroder theorem The combinatorics and Macdonald polynomials The Loehr-Warrington conjecture Solutions to exercises Bibliography.

The q, t-Catalan numbers and the space of diagonal harmonics : with an appendix on the combinatorics of Macdonald polynomials

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive self-adjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some Hopf algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer Hopf algebra of permutations. Among the results surveyed are the Littlewood-Richardson rule and other symmetric function identities, Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, the theory of Lyndon words, the Gessel-Reutenauer bijection, and Hazewinkel's polynomial freeness of QSym. The notes are written with a graduate student reader in mind, being mostly self-contained but requiring a good familiarity with multilinear algebra and -- for the representation-theory applications -- basic group representation theory.

Hopf Algebras in Combinatorics

КРИЗИС ВЛАСТИ В КАБАРДИНО-БАЛКАРИИ В АВГУСТЕ 1991 ГОДА Статья посвящена исследованию влияния путча 1991 г. на этнополитическую обстановку в Кабардино-Балкарской Республике. Анализируется процесс "включения" национального фактора в разгоревшееся политическое противоборство, вылившееся в кризис власти. Авторами впервые сформулирована и обоснована мысль о том, что события августа 1991 г. привели к объединению к тому времени уже нелояльных по отношению друг к другу национальных движений "титульных" народов Кабардино-Балкарии, что нашло выражение в выдвижении общего требования отставки первых лиц республиканской власти. Адрес статьи: www.gramota.net/materials/3/2017/9/58.html

1991년도 사업경과 보고

We want to express digital communication with a human touch. So we pay attention to the properties of a door because it links one place to another. We think the linking of different places and sharing of places is one of the substantial qualities of network technology. Another reason that we focus on a door is because it bears great meanings. For example, a door is the first spot for a meeting with someone or for kissing someone goodbye. It could be your children, spouse, or your friends. These days, we talk and say hello to friends by Messenger or mobiles. Accustomed to digital devices which neglect time or space, we expect that the devices can carry our emotions to others. However, these have limits in expression. We want to show through the our work "The Door" that network technology should have a little more humanity and love in the near future.

The Door

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomials.

Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity

/pdf/a-pieri-rule-for-skew-shapes-142vdhx60j.pdf

A Pieri rule for skew shapes

We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These Kohnert bases provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions.

Kohnert polynomials

Stanley symmetric functions are the stable limits of Schubert polynomials. In this paper, we show that, conversely, Schubert polynomials are Demazure truncations of Stanley symmetric functions. This parallels the relationship between Schur functions and Demazure characters for the general linear group. We establish this connection by imposing a Demazure crystal structure on key tableaux, recently introduced by the first author in connection with Demazure characters and Schubert polynomials, and linking this to the type A crystal structure on reduced word factorizations, recently introduced by Morse and the second author in connection with Stanley symmetric functions.

/pdf/a-demazure-crystal-construction-for-schubert-polynomials-4d7e0fc9m9.pdf

A Demazure crystal construction for Schubert polynomials

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case. Resume. La regle de Pieri exprime le produit d'une fonction de Schur et de la fonction de Schur d'une seule ligne en termes de fonctions de Schur. Nous etendons la regle classique de Pieri en exprimant le produit d'un fonction gauche de Schur et de la fonction de Schur d'une ligne en termes de fonctions gauches de Schur. Comme la regle classique, notre regle implique l'ajout de cases a la forme gauche initiale. Notre preuve est purement combinatoire et etend celle du cas classique.

pdf/a-pieri-rule-for-skew-shapes-3rwvfabrxr.pdf

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Dual equivalence graphs and a combinatorial proof of LLT and Macdonald positivity

A Pieri rule for skew shapes

Kohnert polynomials

A Demazure crystal construction for Schubert polynomials

A Pieri rule for skew shapes