FundamentalBijection

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Contents

• Record Bounds and the Inverse Bijection
• Cycles from Record Blocks
• Partition into Marked Intervals
• An Extreme Value in the Inverse Image
• Separated Prefixes
• Descending Triples and Direct Sums
• Unique Direct-Sum Factorization
• Indecomposability of Pattern Avoiders
• Fixed Avoiders of the Third Iterate
• Boundary Entries of Fixed 231-Avoiders
• The High Prefix before One
• Exclusion of Large Fixed 231-Avoiders
• Low Entries in the Forced Prefix
• Incompatible Middle Cycle Data
• A Consecutive Pair Forced by a Return
• The Prefix before One in a 231-Avoider
• The Penultimate Record Block
• The Penultimate Value of the Original Word
• The Initial Pair after the Maximum and One
• Size Bound for Indecomposable Fixed 312-Avoiders
• The Entry Following the Maximum
• An Upward Edge and a Later Intermediate Value
• Incompatible Terminal Entries
• The Pair before the Terminal Configuration
• The Fundamental Bijection and Fixed Avoiders
• Enumeration of Iterated 132-Avoiders
• A Distinguished Cycle from a Parameter Word
• Reduction along the Distinguished Cycle
• Bounds from the Parameter Cycle
• Decomposition of the Second Avoidance Layer
• Iterated Pattern Avoidance and Its Counts
• Counting Parameters with a Final Maximum
• Shape of Parameters with a Final Maximum
• Location of the Parameter Maximum
• Explicit Exceptional Orbits
• A Descending Parameter at the First Endpoint
• The Final Pair from a Small Second Value
• Classification at the First Endpoint
• The Descending High Prefix
• Insertion of Three Letters
• Cycle Compatibility of Insertion
• Recovering a Smaller Cycle from Insertion
• Cycle Reduction at a Final Maximum
• Recursive Enumeration of First-Endpoint Parameters
• Positional Constraints in Record Blocks
• Record-Block Characterization of 132-Avoidance
• Splitting at the Final Value
• A Cycle Forced by Two Successors
• Appending a Maximal Fixed Point
• Counting Avoiders with a Maximal Tail
• The Increasing Tail Family
• The Descending Parameter Family
• The Balanced Two-Block Family