The periodic (ordinal) patterns of a map are the permutations realized by the relative order of the points in its periodic orbits. We give a combinatorial characterization of the periodic patterns of an arbitrary signed shift, in terms of the structure of the descent set of a certain cyclic permutation associated to the pattern. Signed shifts are an important family of one-dimensional dynamical systems that includes shift maps and the tent map as particular cases. Defined as a function on the set of infinite words on a finite alphabet, a signed shift deletes the first letter and, depending on its value, possibly applies the complementation operation on the remaining word. For shift maps, reverse shift maps, and the tent map, we give exact formulas for their number of periodic patterns. As a byproduct of our work, we recover results of Gessel-Reutenauer and WeissRogers and obtain new enumeration formulas for pattern-avoiding cycles.AMS 2000 subject classifications: Primary 05A15; secondary 37M10, 05A15, 94A55.
We say that an unordered rooted labeled forest avoids the pattern π ∈ Sn if the sequence obtained from the labels along the path from the root to any vertex does not contain a subsequence that is in the same relative order as π. We enumerate several classes of forests that avoid certain sets of permutations, including the set of unimodal forests, via bijections with set partitions with certain properties. We also define and investigate an analog of Wilf-equivalence for forests.
The allowed patterns of a map are those permutations in the same relative order as the initial segments of orbits realized by the map. In this paper, we characterize and provide enumerative bounds for the allowed patterns of signed shifts, a family of maps on infinite words.
An arithmetical structure on a finite, connected graph G is a pair of vectors (d, r) with positive integer entries for which (diag(d) − A)r = 0, where A is the adjacency matrix of G and where the entries of r have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of (diag(d) − A). In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.Let G be a finite, connected graph with n vertices, and let A be the adjacency matrix of G. An arithmetical structure on G is given by a pair of vectors (d, r) ∈ (Z >0 ) n × (Z >0 ) n for which the
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