Exceptional sequences are certain ordered sequences of quiver representations. We introduce a class of objects called strand diagrams and use this model to classify exceptional sequences of representations of a quiver whose underlying graph is a type An Dynkin diagram. We also use variations of this model to classify c-matrices of such quivers, to interpret exceptional sequences as linear extensions of posets, and to give a simple bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This work is an extension of the classification of exceptional sequences for the linearly-ordered quiver by the first and third authors. ‚ chains in the lattice of noncrossing partitions [Bes03, HK13, IT09], ‚ c-matrices and cluster algebras [ST13], ‚ factorizations of Coxeter elements [IS10], and ‚ t-structures and derived categories [Bez03, BK89, Rud90].Despite their ubiquity, very little work has been done to concretely describe exceptional sequences, even for path algebras of Dynkin quivers [Ara13, GM15]. In this paper, we give a concrete description of exceptional sequences
We develop basic cluster theory from an elementary point of view using a variation of binary trees which we call mixed cobinary trees. We show that the number of isomorphism classes of such trees is given by the Catalan number C n where n is the number of internal nodes. We also consider the corresponding quiver Q ǫ of type A n−1 . As a special case of more general known results about the relation between c-vectors, representations of quivers and their semi-invariants, we explain the bijection between mixed cobinary trees and the vertices of the generalized associahedron corresponding to the quiver Q ǫ .
Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D 4 , the symmetry group of a square. We examine pattern-avoiding permutations with 180 • rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our results involve well-known sequences such as the alternating Fibonacci numbers, Catalan numbers, triangular numbers, and powers of two.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.