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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