Consecutive patterns and statistics on restricted permutations

El tema d'aquesta tesi es l'enumeracio de permutacions amb subsequencies prohibides respecte a certs estadistics, i l'enumeracio de permutacions que eviten subsequencies generalitzades. Despres d'introduir algunes definicions sobre subsequencies i estadistics en permutacions i camins de Dyck, comencem estudiant la distribucio dels estadistics -nombre de punts fixos' i -nombre d'excedencies' en permutacions que eviten una subsequencia de longitud 3. Un dels resultats principals es que la distribucio conjunta d'aquest parell de parametres es la mateixa en permutacions que eviten 321 que en permutacions que eviten 132. Aixo generalitza un teorema recent de Robertson, Saracino i Zeilberger. Demostrem aquest resultat donant una bijeccio que preserva els dos estadistics en questio i un altre parametre. La idea clau consisteix en introduir una nova classe d'estadistics en camins de Dyck, basada en el que anomenem tunel. A continuacio considerem el mateix parell d'estadistics en permutacions que eviten simultaniament dues o mes subsequencies de longitud 3. Resolem tots els casos donant les funcions generadores corresponents. Alguns casos son generalitzats a subsequencies de longitud arbitraria. Tambe descrivim la distribucio d'aquests parametres en involucions que eviten qualsevol subconjunt de subsequencies de longitud 3. La tecnica principal consisteix en fer servir bijeccions entre permutacions amb subsequencies prohibides i certs tipus de camins de Dyck, de manera que els estadistics en permutacions que considerem corresponen a estadistics en camins de Dyck que son mes facils d'enumerar. Tot seguit presentem una nova familia de bijeccions del conjunt de camins de Dyck a si mateix, que envien estadistics que apareixen en l'estudi de permutacions amb subsequencies prohibides a estadistics classics en camins de Dyck, la distribucio dels quals s'obte facilment. En particular, aixo ens dona una prova bijectiva senzilla de l'equidistribucio de punts fixos en les permutacions que eviten 321 i en les que eviten 132. A continuacio donem noves interpretacions dels nombres de Catalan i dels nombres de Fine. Considerem una classe de permutacions definida en termes d'aparellaments de 2n punts en una circumferencia sense creuaments. N'estudiem l'estructura i algunes propietats, i donem la distribucio de diversos estadistics en aquests permutacions. En la seguent part de la tesi introduim una nocio diferent de subsequencies prohibides, amb el requeriment que els elements que formen la subsequencia han d'apareixer en posicions consecutives a la permutacio. Mes en general, estudiem la distribucio del nombre d'ocurrencies de subparaules (subsequencies consecutives) en permutacions. Resolem el problema en diversos casos segons la forma de la subparaula, obtenint-ne les funcions generadores exponencials bivariades corresponents com a solucions de certes equacions diferencials lineals. El metode esta basat en la representacio de permutacions com a arbres binaris creixents i en metodes simbolics. La part final tracta de subsequencies generalitzades, que extenen tant la nocio de subsequencies classiques com la de subparaules. Per algunes subsequencies obtenim nous resultats enumeratius. Finalment estudiem el comportament assimptotic del nombre de permutacions de mida n que eviten una subsequencia generalitzada fixa quan n tendeix a infinit. Tambe donem fites inferiors i superiors en el nombre de permutacions que eviten certes subsequencies.

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