Permuting machines

Mike Atkinson

A hole in the ground

45 3 2 1

A line of golf balls about to fall into a hole

A hole in the ground

4

53

2

1

They trickled down on both sides of the dividing rock

An input restricted deque

45 3 2 1

Input is allowed into one end of a linear list but output is allowed from both ends

An input restricted deque

4 5 32 1

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

3

2 1

2 1

2

12

1

3

2

13 2

132 1 3

21 3

21

Can produce

Can produce

or

But not321

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 cn of permutations of length n in the ideal is at most kn

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 functionsPolynomial

Rational

Algebraic

Wild