Equivalence Classes of Permutations Modulo Replacements Between 123 and Two-Integer Patterns

We explore a new type of replacement of patterns in permutations, suggested by James Propp, that does not preserve the length of permutations. In particular, we focus on replacements between 123 and a pattern of two integer elements. We apply these replacements in the classical sense; that is, the elements being replaced need not be adjacent in position or value. Given each replacement, the set of all permutations is partitioned into equivalence classes consisting of permutations reachable from one another through a series of bi-directional replacements. We break the eighteen replacements of interest into four categories by the structure of their classes and fully characterize all of their classes.Comment: 14 page

Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previously unsolved case. When the partition is of $S_3$ and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for $13$ of the $14$ unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure