Open Meanders

Overview

Open meanders help to identify the various combinations of circuits that are viable as unicursal paths for uniaxial labyrinths or components of more complex labyrinths. The enumeration of open meanders is a well-established mathematical problem. However, it is difficult to find a list of the individual sequences, so I have been identifying viable permutations [update: the Combinatorial Object Server has been relaunched!]. Part of the challenge is how quickly the numbers increase with each additional circuit added (see Meandric Numbers tab below).

David Bevan’s Open Meanders Wolfram Demonstrations Project helps to visualize many of the open meanders. Tony Phillips' Through Mazes to Mathematics web pages include about 50 level sequences and 30 level diagrams. Andreas Frei's and Erwin Reißmann's blogmymaze posts show some of the open meanders with their corresponding labyrinths. I have added over 2,000 open meander sequences so far to the spreadsheet below (see OM Viable Permutations tab) as well as all open meander symmetry groups through 11 circuits  (see OM Groups tab).

While I have joked about wanting to know ALL of the labyrinths, the real goal here is to analyze enough of them to understand and describe recurring component patterns and relationships among labyrinths, as well as to number enough of the open meanders (SAT labyrinths) to inform analysis of divided and complex labyrinths. I have organized them in numerical order by circuit sequence and numbered them within each number of circuits. The naming convention is "number of circuits.sequence number" (e.g., OM 5.6 is the 6th 5-circuit sequence, 52341, the double meander). 

For all open meanders, the first sequence is always from the serpentine series (0.1, 1.1, 2.1, 3.1, etc.). For all odd numbers of circuits, the last sequence is always from the reverse serpentine series (1.1, 3.2, 5.8, 7.42, etc.).

Open Meanders and Symmetry Groups Dataset