13,055 research outputs found
On the X-rays of permutations
The X-ray of a permutation is defined as the sequence of antidiagonal sums in
the associated permutation matrix. X-rays of permutation are interesting in the
context of Discrete Tomography since many types of integral matrices can be
written as linear combinations of permutation matrices. This paper is an
invitation to the study of X-rays of permutations from a combinatorial point of
view. We present connections between these objects and nondecreasing
differences of permutations, zero-sum arrays, decomposable permutations, score
sequences of tournaments, queens' problems and rooks' problems.Comment: 7 page
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
Cambrian Lattices
For an arbitrary finite Coxeter group W we define the family of Cambrian
lattices for W as quotients of the weak order on W with respect to certain
lattice congruences. We associate to each Cambrian lattice a complete fan,
which we conjecture is the normal fan of a polytope combinatorially isomorphic
to the generalized associahedron for W. In types A and B we obtain, by means of
a fiber-polytope construction, combinatorial realizations of the Cambrian
lattices in terms of triangulations and in terms of permutations. Using this
combinatorial information, we prove in types A and B that the Cambrian fans are
combinatorially isomorphic to the normal fans of the generalized associahedra
and that one of the Cambrian fans is linearly isomorphic to Fomin and
Zelevinsky's construction of the normal fan as a "cluster fan." Our
construction does not require a crystallographic Coxeter group and therefore
suggests a definition, at least on the level of cellular spheres, of a
generalized associahedron for any finite Coxeter group. The Tamari lattice is
one of the Cambrian lattices of type A, and two "Tamari" lattices in type B are
identified and characterized in terms of signed pattern avoidance. We also show
that open intervals in Cambrian lattices are either contractible or homotopy
equivalent to spheres.Comment: Revisions in exposition (partly in response to the suggestions of an
anonymous referee) including many new figures. Also, Conjecture 1.4 and
Theorem 1.5 are replaced by slightly more detailed statements. To appear in
Adv. Math. 37 pages, 8 figure
Symmetries and Paraparticles as a Motivation for Structuralism
This paper develops an analogy proposed by Stachel between general relativity
(GR) and quantum mechanics (QM) as regards permutation invariance. Our main
idea is to overcome Pooley's criticism of the analogy by appeal to
paraparticles.
In GR the equations are (the solution space is) invariant under
diffeomorphisms permuting spacetime points. Similarly, in QM the equations are
invariant under particle permutations. Stachel argued that this feature--a
theory's `not caring which point, or particle, is which'--supported a
structuralist ontology.
Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions
and bosons implies that each individual state (solution) is fixed by each
permutation, while in GR a diffeomorphism yields in general a distinct, albeit
isomorphic, solution.
We define various versions of structuralism, and go on to formulate Stachel's
and Pooley's positions, admittedly in our own terms. We then reply to Pooley.
Though he is right about fermions and bosons, QM equally allows more general
types of symmetry, in which states (vectors, rays or density operators) are not
fixed by all permutations (called `paraparticle states'). Thus Stachel's
analogy is revived.Comment: 45 pages, Latex, 3 Figures; forthcoming in British Journal for the
Philosophy of Scienc
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