123 research outputs found
Dynamical aspects of -machines
The -machine was recently introduced by Cerbai, Claesson and Ferrari
as a tool to gain a better insight on the problem of sorting permutations with
two stacks in series. It consists of two consecutive stacks, which are
restricted in the sense that their content must at all times avoid a certain
pattern: a given , in the first stack, and , in the second. Here we
prove that in most cases sortable permutations avoid a bivincular pattern
. We provide a geometric decomposition of -avoiding permutations and
use it to count them directly. Then we characterize the permutations with the
property that the output of the -avoiding stack does not contain
, which we call effective. For , we obtain an alternative
method to enumerate sortable permutations. Finally, we classify
-machines and determine the most challenging to be studied.Comment: 19 pages, 4 figures, 3 tables. arXiv admin note: text overlap with
arXiv:2210.0362
Stack sorting with restricted stacks
The (classical) problem of characterizing and enumerating permutations that
can be sorted using two stacks connected in series is still largely open. In
the present paper we address a related problem, in which we impose restrictions
both on the procedure and on the stacks. More precisely, we consider a greedy
algorithm where we perform the rightmost legal operation (here "rightmost"
refers to the usual representation of stack sorting problems). Moreover, the
first stack is required to be -avoiding, for some permutation ,
meaning that, at each step, the elements maintained in the stack avoid the
pattern when read from top to bottom. Since the set of permutations
which can be sorted by such a device (which we call -machine) is not
always a class, it would be interesting to understand when it happens. We will
prove that the set of -machines whose associated sortable permutations
are not a class is counted by Catalan numbers. Moreover, we will analyze two
specific -machines in full details (namely when and
), providing for each of them a complete characterization and
enumeration of sortable permutations
2-stack pushall sortable permutations
In the 60's, Knuth introduced stack-sorting and serial compositions of
stacks. In particular, one significant question arise out of the work of Knuth:
how to decide efficiently if a given permutation is sortable with 2 stacks in
series? Whether this problem is polynomial or NP-complete is still unanswered
yet. In this article we introduce 2-stack pushall permutations which form a
subclass of 2-stack sortable permutations and show that these two classes are
closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a
given permutation of size n is 2-stack pushall sortable and describe all its
sortings. This result is a step to the solve the general 2-stack sorting
problem in polynomial time.Comment: 41 page
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