Permuting machines

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Permuting machines

• Mike Atkinson

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A hole in the ground
5
4
3
2
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A line of golf balls about to fall into a hole
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A hole in the ground
They trickled down on both sides of the dividing
rock
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An input restricted deque
Input is allowed into one end of a linear list
but output is allowed from both ends
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An input restricted deque
A possible output order
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An exclusive art gallery
Exit enlightened
Pay and enter
Four paintings in two very small rooms
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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

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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 doesnt solve the problems in every case

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Back to the hole in the ground
Can produce
or
Can produce
But not
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Hole in the ground permutations
If c gt b gt a such a permutation cannot be
generated
If there is no such c gt b gt a the permutation
can be generated
321 is the characterising forbidden subpermutation
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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

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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

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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 dont
  remember it - but the generating function is
  rational

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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?

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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