6 research outputs found
Short Proofs for Cut-and-Paste Sorting of Permutations
We consider the problem of determining the maximum number of moves required
to sort a permutation of using cut-and-paste operations, in which a
segment is cut out and then pasted into the remaining string, possibly
reversed. We give short proofs that every permutation of can be
transformed to the identity in at most \flr{2n/3} such moves and that some
permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure
On the effective and automatic enumeration of polynomial permutation classes
We describe an algorithm, implemented in Python, which can enumerate any
permutation class with polynomial enumeration from a structural description of
the class. In particular, this allows us to find formulas for the number of
permutations of length n which can be obtained by a finite number of block
sorting operations (e.g., reversals, block transpositions, cut-and-paste
moves)
Sorting by shuffling methods and a queue
We consider sorting by a queue that can apply a permutation from a given set
over its content. This gives us a sorting device
corresponding to any shuffling method since every such method is
associated with a set of permutations. Two variations of these devices are
considered - and
. These require the entire content of the
device to be unloaded after a permutation is applied or unloaded by each pop
operation, respectively.
First, we show that sorting by a deque is equivalent to sorting by a queue
that can reverse its content. Next, we focus on sorting by cuts. We prove that
the set of permutations that one can sort by using
is the set of the -avoiding separable
permutations. We give lower and upper bounds to the maximum number of times the
device must be used to sort a permutation. Furthermore, we give a formula for
the number of -permutations, , that one
can sort by using , for any shuffling method
, such that the permutations associated with it are irreducible.
Next, we prove a generalization of the fact that
can sort all permutations. We also show
that is given by the odd indexed
Fibonacci numbers , for any shuffling method having a
specific back-front property. The rest of the work is dedicated to a surprising
conjecture inspired by Diaconis and Graham which states that one can sort the
same number of permutations of any given size by using the devices
and
, corresponding to the popular
In-shuffle and Monge shuffling methods.Comment: 29 pages, 7 figure