Animations

The interactive pages of the research projects, each running in the browser.

• Schubert varieties and Schubert polynomials

  

  Sets of permutations from Schubert varieties and Schubert polynomials, most of them permutation classes with a combinatorial specification. Type a permutation to see which of them it belongs to, where it sits among them, and how each specification builds it.

• Sorting

  

  The permutations that sorting machines sort: stacks, queues, pop-stacks and deques, alone, in series or in parallel, and sorting maps such as West's stack-sorting map and bubble sort. Type a permutation to see which machines sort it, watch them at work, see where it sits among the classes, and how each specification builds it.

• How TileScope expands a class

  

  TileScope looks for a combinatorial specification by expanding tilings. Each expansion finds rules: a tiling written in terms of other tilings. The tilings and rules found so far are the search's universe, and every so often comb_spec_searcher checks it for a specification. Here each class's search is replayed as it was recorded, tiling by tiling between the checks.

• Uniform random generation

  

  A combinatorial specification counts its class, and the counts are all it takes to draw a permutation uniformly at random. Going down the specification, the draw takes each branch, and each way of splitting its points, in proportion to how many permutations lie that way. Watch one draw make its choices and build its permutation, then let thousands of draws gather into a heatmap of the class.

• Counting the permutation classes with bases of patterns of length 4 and 5

  

  Every permutation class whose basis is a set of patterns of length 4, almost 17 million of them, cut down step by step: up to symmetry, then the classes with a regular insertion encoding, then those with finitely many simples. Combinatorial Exploration found a specification for every class left after the insertion encoding except Av(1324) and its reverse. The same steps are then counted for bases with patterns of lengths 4 and 5.

Their animations also play one after another in a full-screen show, as the papers' figures do in a slideshow.