181 research outputs found
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
Ascent sequences and upper triangular matrices containing non-negative integers
This paper presents a bijection between ascent sequences and upper triangular
matrices whose non-negative entries are such that all rows and columns contain
at least one non-zero entry. We show the equivalence of several natural
statistics on these structures under this bijection and prove that some of
these statistics are equidistributed. Several special classes of matrices are
shown to have simple formulations in terms of ascent sequences. Binary matrices
are shown to correspond to ascent sequences with no two adjacent entries the
same. Bidiagonal matrices are shown to be related to order-consecutive set
partitions and a simple condition on the ascent sequences generate this class.Comment: 13 page
The area above the Dyck path of a permutation
In this paper we study a mapping from permutations to Dyck paths. A Dyck path
gives rise to a (Young) diagram and we give relationships between statistics on
permutations and statistics on their corresponding diagrams. The distribution
of the size of this diagram is discussed and a generalisation given of a parity
result due to Simion and Schmidt. We propose a filling of the diagram which
determines the permutation uniquely. Diagram containment on a restricted class
of permutations is shown to be related to the strong Bruhat poset.Comment: 9 page
Permutations sortable by n-4 passes through a stack
We characterise and enumerate permutations that are sortable by n-4 passes
through a stack. We conjecture the number of permutations sortable by n-5
passes, and also the form of a formula for the general case n-k, which involves
a polynomial expression.Comment: 6 page
An Ising model having permutation spin motivated by a permutation complexity measure
In this paper we define a variant of the Ising model in which spins are
replaced with permutations. The energy between two spins is a function of the
relative disorder of one spin, a permutation, to the other. This model is
motivated by a complexity measure for declarative systems. For such systems a
state is a permutation and the permutation sorting complexity measures the
average sequential disorder of neighbouring states. To measure the relative
disorder between two spins we use a symmetrized version of the descent
permutation statistic that has appeared in the works of Chatterjee \& Diaconis
and Petersen. The classical Ising model corresponds to the length-2 permutation
case of this new model. We consider and prove some elementary properties for
the 1D case of this model in which spins are length-3 permutations.Comment: Dedicated to Einar Steingr\'imsson on the occasion of his retirement.
15 pages, 2 figure
The Federal Disaster Assistance Policy -- a declarative analysis
In this paper we will provide a quantitative analysis of the Federal Disaster
Assistance policy from the viewpoint of three different stakeholders. This
quantitative methodology is new and has applications to other areas such as
business and healthcare processes. The stakeholders are interested in process
transparency but each has a different opinion on precisely what constitutes
transparency. We will also consider three modifications to the Federal Disaster
Assistance policy and analyse, from a stakeholder viewpoint, how stakeholder
satisfaction changes from process to process. This analysis is used to rank the
favourability of four policies with respect to all collective stakeholder
preferences
Generalized ballot sequences are ascent sequences
Ascent sequences were introduced by the author (in conjunction with others) to encode a class of permutations that avoid a single length- three bivincular pattern, and were the central object through which other combinatorial correspondences were discovered. In this note we prove the non-trivial fact that generalized ballot sequences are ascent sequences
Enumerating (2+2)-free posets by indistinguishable elements
A poset is said to be (2+2)-free if it does not contain an induced subposet
that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two
elements in a poset are indistinguishable if they have the same strict up-set
and the same strict down-set. Being indistinguishable defines an equivalence
relation on the elements of the poset. We introduce the statistic maxindist,
the maximum size of a set of indistinguishable elements. We show that, under a
bijection of Bousquet-Melou et al., indistinguishable elements correspond to
letters that belong to the same run in the so-called ascent sequence
corresponding to the poset. We derive the generating function for the number of
(2+2)-free posets with respect to both maxindist and the number of different
strict down-sets of elements in the poset. Moreover, we show that (2+2)-free
posets P with maxindist(P) at most k are in bijection with upper triangular
matrices of nonnegative integers not exceeding k, where each row and each
column contains a nonzero entry. (Here we consider isomorphic posets to be
equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1
correspond to upper triangular binary matrices where each row and column
contains a nonzero entry, and whose entries sum to n. We derive a generating
function counting such matrices, which confirms a conjecture of Jovovic, and we
refine the generating function to count upper triangular matrices consisting of
nonnegative integers not exceeding k and having a nonzero entry in each row and
column. That refined generating function also enumerates (2+2)-free posets
according to maxindist. Finally, we link our enumerative results to certain
restricted permutations and matrices.Comment: 16 page
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