ﻻ يوجد ملخص باللغة العربية
We introduce $delta$-cliffs, a generalization of permutations and increasing trees depending on a range map $delta$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $delta$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak order of permutations, we construct algebras whose products form intervals of the lattices of $delta$-cliff. We provide necessary and sufficient conditions on $delta$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $delta$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra and we get new generalizations of this structure.
Three families of posets depending on a nonnegative integer parameter $m$ are introduced. The underlying sets of these posets are enumerated by the $m$-Fuss Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is
A combinatorial structure, $mathcal{F}$, with counting sequence ${a_n}_{nge 0}$ and ordinary generating function $G_mathcal{F}=sum_{nge0} a_n x^n$, is positive algebraic if $G_mathcal{F}$ satisfies a polynomial equation $G_mathcal{F}=sum_{k=0}^N p_k(
Squared singular values of a product of s square random Ginibre matrices are asymptotically characterized by probability distribution P_s(x), such that their moments are equal to the Fuss-Catalan numbers or order s. We find a representation of the Fu
Motivated by generalizing Khovanovs categorification of the Jones polynomial, we study functors $F$ from thin posets $P$ to abelian categories $mathcal{A}$. Such functors $F$ produce cohomology theories $H^*(P,mathcal{A},F)$. We find that CW posets,
We classify, up to isomorphism, the 2-dimensional algebras over a field K. We focuse also on the case of characteristic 2, identifying the matrices of GL(2,F_2) with the elements of the symmetric group S_3. The classification is then given by the stu