Permuting machines Mike Atkinson

A hole in the ground A line of golf balls about to fall into a hole

A hole in the ground They trickled down on both sides of the dividing rock

An input restricted deque Input is allowed into one end of a linear list but output is allowed from both ends

An input restricted deque A possible output order

An exclusive art gallery Pay and enter Exit enlightened Four paintings in two very small rooms

Permuting machines All the examples have an “output” that is a permutation of the input So they are associated with a certain set of permutations that represent the computations they can do Other examples: container data structures, packet-switching networks, sorting by imperfect algorithms

Permuting machines Can we characterise the possible permutations of a permuting machine? Can we enumerate them for each fixed length? Under mild conditions a general theory can be built But it doesn’t solve the problems in every case

Back to the hole in the ground Can produce or But not 3 2 1

Hole in the ground permutations c a b If c > b > a such a permutation cannot be generated If there is no such c > b > a the permutation can be generated 321 is the characterising forbidden subpermutation

Forbidden subpermutations Hole in the ground permutations are exactly those that do not have 321 as a subpermutation Restricted deque permutations are exactly those that do not have 4213 and 4231 as a subpermutation Art gallery permutations are characterised in a similar way but we need infinitely many forbidden subpermutations

Forbidden subpermutations Many permuting machines have their permutations defined by a list (often an infinite list) of forbidden subpermutations Such permutation sets are precisely those that are ideals in the “subpermutation order” These ideals might be compared to the ideals for the “graph minor” order but they are more complicated

Counting The number of hole in the ground permutations of length n is The number of restricted deque permutations of length n are the coefficients in The number of art gallery permutations of length n is known, even more complicated, and I don’t remember it - but the generating function is rational

Milestone result and problem Marcus & Tardos (2004): for any proper ideal there is a constant k for which the number c n of permutations of length n in the ideal is at most k n Is it true that exists?

General questions Given an ideal in the subpermutation order find a list of forbidden permutations that characterises it, and Determine the number of permutations in the ideal of each length n Understand the counting functions Polynomial Rational Algebraic Wild