6,942 research outputs found
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
Actions on permutations and unimodality of descent polynomials
We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis ,
. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns and are invariant under the action and use this to
prove unimodality properties for a -analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the -Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.Comment: 19 pages, revised version to appear in Europ. J. Combi
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting
Type A affine shuffles are compared with riffle shuffles followed by a cut.
Although these probability measures on the symmetric group S_n are different,
they both satisfy a convolution property. Strong evidence is given that when
the underlying parameter satisfies , the induced measures on
conjugacy classes of the symmetric group coincide. This gives rise to
interesting combinatorics concerning the modular equidistribution by major
index of permutations in a given conjugacy class and with a given number of
cyclic descents. It is proved that the use of cuts does not speed up the
convergence rate of riffle shuffles to randomness. Generating functions for the
first pile size in patience sorting from decks with repeated values are
derived. This relates to random matrices.Comment: Galley version for J. Alg.; minor revisions in Sec.
Enumeration of Stack-Sorting Preimages via a Decomposition Lemma
We give three applications of a recently-proven "Decomposition Lemma," which
allows one to count preimages of certain sets of permutations under West's
stack-sorting map . We first enumerate the permutation class
, finding a new example
of an unbalanced Wilf equivalence. This result is equivalent to the enumeration
of permutations sortable by , where is the bubble
sort map. We then prove that the sets ,
,
and are
counted by the so-called "Boolean-Catalan numbers," settling a conjecture of
the current author and another conjecture of Hossain. This completes the
enumerations of all sets of the form
for
with the exception of the set
. We also find an explicit formula for
, where
is the set of permutations in with descents.
This allows us to prove a conjectured identity involving Catalan numbers and
order ideals in Young's lattice.Comment: 20 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1903.0913
Revstack sort, zigzag patterns, descent polynomials of -revstack sortable permutations, and Steingr\'imsson's sorting conjecture
In this paper we examine the sorting operator . Applying
this operator to a permutation is equivalent to passing the permutation
reversed through a stack. We prove theorems that characterise -revstack
sortability in terms of patterns in a permutation that we call
patterns. Using these theorems we characterise those permutations of length
which are sorted by applications of for . We
derive expressions for the descent polynomials of these six classes of
permutations and use this information to prove Steingr\'imsson's sorting
conjecture for those six values of . Symmetry and unimodality of the descent
polynomials for general -revstack sortable permutations is also proven and
three conjectures are given
Two Vignettes On Full Rook Placements
Using bijections between pattern-avoiding permutations and certain full rook
placements on Ferrers boards, we give short proofs of two enumerative results.
The first is a simplified enumeration of the 3124, 1234-avoiding permutations,
obtained recently by Callan via a complicated decomposition. The second is a
streamlined bijection between 1342-avoiding permutations and permutations which
can be sorted by two increasing stacks in series, originally due to Atkinson,
Murphy, and Ru\v{s}kuc.Comment: 9 pages, 4 figure
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