3,944 research outputs found
Shuffle operations on discrete paths
AbstractWe consider the shuffle operation on paths and study some parameters. In the case of square lattices, shuffling with a particular periodic word (of period 2) corresponding to paperfoldings reveals some characteristic properties: closed paths remain closed; the area and perimeter double; the center of gravity moves under a 45∘ rotation and a 2 zoom factor. We also observe invariance properties for the associated Dragon curves. Moreover, replacing square lattice paths by paths involving 2kπ/N-turns, we find analogous results using more general shuffles
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
An algorithm to prescribe the configuration of a finite graph
We provide algorithms involving edge slides, for a connected simple graph to
evolve in a finite number of steps to another connected simple graph in a
prescribed configuration, and for the regularization of such a graph by the
minimization of an appropriate energy functional
Cohomology of idempotent braidings, with applications to factorizable monoids
We develop new methods for computing the Hochschild (co)homology of monoids
which can be presented as the structure monoids of idempotent set-theoretic
solutions to the Yang--Baxter equation. These include free and symmetric
monoids; factorizable monoids, for which we find a generalization of the
K{\"u}nneth formula for direct products; and plactic monoids. Our key result is
an identification of the (co)homologies in question with those of the
underlying YBE solutions, via the explicit quantum symmetrizer map. This
partially answers questions of Farinati--Garc{\'i}a-Galofre and Dilian Yang. We
also obtain new structural results on the (co)homology of general YBE
solutions
Combinatorial Hopf algebra structure on packed square matrices
We construct a new bigraded Hopf algebra whose bases are indexed by square
matrices with entries in the alphabet , , without
null rows or columns. This Hopf algebra generalizes the one of permutations of
Malvenuto and Reutenauer, the one of -colored permutations of Novelli and
Thibon, and the one of uniform block permutations of Aguiar and Orellana. We
study the algebraic structure of our Hopf algebra and show, by exhibiting
multiplicative bases, that it is free. We moreover show that it is self-dual
and admits a bidendriform bialgebra structure. Besides, as a Hopf subalgebra,
we obtain a new one indexed by alternating sign matrices. We study some of its
properties and algebraic quotients defined through alternating sign matrices
statistics.Comment: 35 page
Quillen homology for operads via Gr\"obner bases
The main goal of this paper is to present a way to compute Quillen homology
of operads. The key idea is to use the notion of a shuffle operad we introduced
earlier; this allows to compute, for a symmetric operad, the homology classes
and the shape of the differential in its minimal model, although does not give
an insight on the symmetric groups action on the homology. Our approach goes in
several steps. First, we regard our symmetric operad as a shuffle operad, which
allows to compute its Gr\"obner basis. Next, we define a combinatorial
resolution for the "monomial replacement" of each shuffle operad (provided by
the Gr\"obner bases theory). Finally, we explain how to "deform" the
differential to handle every operad with a Gr\"obner basis, and find explicit
representatives of Quillen homology classes for a large class of operads. We
also present various applications, including a new proof of Hoffbeck's PBW
criterion, a proof of Koszulness for a class of operads coming from commutative
algebras, and a homology computation for the operads of Batalin-Vilkovisky
algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint
arXiv:0912.4895. Final version, to appear in Documenta Mat
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