Sample programs

The programs here are written in C++, using the CWEBx system of software documentation. Links are provided to the tangled (directly compilable) sources and the generated documentation in PDF format, so to use and inspect the programs it is not necessary to install the CWEBx system.

Combing lattice paths to obtain tilings of the Aztec diamond

CWEBx source file aztec.w
Tangled source file aztec.cpp
Woven documentation aztec.pdf

This program will apply the algorithm described in the paper "A bijection proving the Aztec diamond theorem by combing lattice paths", either on a random array of bits to generate a random path family, or exhaustively on all possible such data of a given order; it also runs the reverse algorithm checking that it recovers the initial data. It can produce PostScript data depicting the path family produced, and optionally several intermediate stages, and the corresponding Aztec diamond tiling. The program is a command line program taking its arguments from the command line. Assuming the executable program is named aztec, the calling syntax is

  aztec [OPTIONS] ORDER [FILE]

where ORDER is the order of the Aztec diamond to consider (a nonnegative number; the path family will contain one more path than this). The final argument FILE is a base for the name of the PostScript file to be produced; the program will automatically add "-ORDER.ps" to it. If no FILE argument is given but PostScript output is requested, it will be written to the standard output. The default is to output one randomly generated path family of the given order, but this can be modified by options. The possible options are

-x Do exhaustive generation in given order, while suppressing PostScript output; just report success or failure of executing the algorithm and inverting it
-X Do exhaustive generation, and produce (potentially massive) PostScript output
-t Produce (also) output of the completed configuration as a tiling of the Aztec diamond
-n Suppress output of a path family
-s NUM Write, to the PostScript output, NUM intermediate snapshots during the algorithm, done at more or less equally spaced times when complete columns have been made disjoint, the first snapshot being at the very beginning. Setting this option also causes "-NUM" to be inserted into the file name produced, just before the dot. An alternative syntax for setting this option is to specify ORDER in the form n/s where n is the order and s the number of snapshots.
-c NUM Write, to the PostScript output, in detail the stages during which column number NUM is made disjoint and obtains its final value; this happens after the path with the same number is taken into consideration.
-r NUM Set the random seed to NUM, so that reproducible output can be generated.

A bijection between partitions of n and strict partitions of 2n whose diagram is black/white balanced

This is a small Python program that computes the bijections described in these MathOverflow and Math.StackExchange answers. They are between partitions of n on one hand and on the other hand strict partitions of 2n with empty 2-core, where the latter means that the diagram when coloured in a checkerboard fashion has as many black as white squares, or equivalently that the odd parts of the partition (which are even in number) occupy as many odd as even positions.

Python source file blowup.py

This is my very first Python program, so it might contain things that are frowned upon by the Python community.

The forward direction is computed by double_strict using edge sequences, and also by to_double using the more direct approach of the StackExchange answer (the two functions should produce the same results). Similarly the reverse direction (which is actually the one described in the answers) is computed by undouble using edge sequences, and also more directly by by to_plain.