73,715 research outputs found
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
Constructions for cyclic sieving phenomena
We show how to derive new instances of the cyclic sieving phenomenon from old
ones via elementary representation theory. Examples are given involving objects
such as words, parking functions, finite fields, and graphs.Comment: 18 pages, typos fixed, to appear in SIAM J. Discrete Mat
Consistency of Maximum Likelihood for Continuous-Space Network Models
Network analysis needs tools to infer distributions over graphs of arbitrary
size from a single graph. Assuming the distribution is generated by a
continuous latent space model which obeys certain natural symmetry and
smoothness properties, we establish three levels of consistency for
non-parametric maximum likelihood inference as the number of nodes grows: (i)
the estimated locations of all nodes converge in probability on their true
locations; (ii) the distribution over locations in the latent space converges
on the true distribution; and (iii) the distribution over graphs of arbitrary
size converges.Comment: 21 page
Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories
A decorated surface S is an oriented surface with punctures and a finite set
of marked points on the boundary, such that each boundary component has a
marked point. We introduce ideal bipartite graphs on S. Each of them is related
to a group G of type A, and gives rise to cluster coordinate systems on certain
spaces of G-local systems on S. These coordinate systems generalize the ones
assigned to ideal triangulations of S. A bipartite graph on S gives rise to a
quiver with a canonical potential. The latter determines a triangulated 3d CY
category with a cluster collection of spherical objects. Given an ideal
bipartite graph on S, we define an extension of the mapping class group of S
which acts by symmetries of the category. There is a family of open CY 3-folds
over the universal Hitchin base, whose intermediate Jacobians describe the
Hitchin system. We conjecture that the 3d CY category with cluster collection
is equivalent to a full subcategory of the Fukaya category of a generic
threefold of the family, equipped with a cluster collection of special
Lagrangian spheres. For SL(2) a substantial part of the story is already known
thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others. We
hope that ideal bipartite graphs provide special examples of the
Gaiotto-Moore-Neitzke spectral networks.Comment: 60 page
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