1,749 research outputs found
Overpartitions, lattice paths and Rogers-Ramanujan identities
We extend partition-theoretic work of Andrews, Bressoud, and Burge to
overpartitions, defining the notions of successive ranks, generalized Durfee
squares, and generalized lattice paths, and then relating these to
overpartitions defined by multiplicity conditions on the parts. This leads to
many new partition and overpartition identities, and provides a unification of
a number of well-known identities of the Rogers-Ramanujan type. Among these are
Gordon's generalization of the Rogers-Ramanujan identities, Andrews'
generalization of the G\"ollnitz-Gordon identities, and Lovejoy's ``Gordon's
theorems for overpartitions.
Updown categories: Generating functions and universal covers
A poset can be regarded as a category in which there is at most one morphism
between objects, and such that at most one of Hom(c,c') and Hom(c',c) is
nonempty for distinct objects c,c'. If we keep in place the latter axiom but
allow for more than one morphism between objects, we have a sort of generalized
poset in which there are multiplicities attached to covering relations, and
possibly nontrivial automorphism groups. We call such a category an "updown
category". In this paper we give a precise definition of such categories and
develop a theory for them. We also give a detailed account of ten examples,
including updown categories of integer partitions, integer compositions, planar
rooted trees, and rooted trees.Comment: arXiv admin note: substantial text overlap with arXiv:math/040245
The toggle group, homomesy, and the Razumov-Stroganov correspondence
The Razumov-Stroganov correspondence, an important link between statistical
physics and combinatorics proved in 2011 by L. Cantini and A. Sportiello,
relates the ground state eigenvector of the O(1) dense loop model on a
semi-infinite cylinder to a refined enumeration of fully-packed loops, which
are in bijection with alternating sign matrices. This paper reformulates a key
component of this proof in terms of posets, the toggle group, and homomesy, and
proves two new homomesy results on general posets which we hope will have
broader implications.Comment: 14 pages, 10 figures, final versio
Rational Dyck Paths in the Non Relatively Prime Case
We study the relationship between rational slope Dyck paths and invariant
subsets of extending the work of the first two authors in the
relatively prime case. We also find a bijection between --Dyck paths
and -tuples of -Dyck paths endowed with certain gluing data. These
are the first steps towards understanding the relationship between rational
slope Catalan combinatorics and the geometry of affine Springer fibers and knot
invariants in the non relatively prime case.Comment: 25 pages, 9 figure
A second look at the toric h-polynomial of a cubical complex
We provide an explicit formula for the toric -contribution of each cubical
shelling component, and a new combinatorial model to prove Clara Chan's result
on the non-negativity of these contributions. Our model allows for a variant of
the Gessel-Shapiro result on the -polynomial of the cubical lattice, this
variant may be shown by simple inclusion-exclusion. We establish an isomorphism
between our model and Chan's model and provide a reinterpretation in terms of
noncrossing partitions. By discovering another variant of the Gessel-Shapiro
result in the work of Denise and Simion, we find evidence that the toric
-polynomials of cubes are related to the Morgan-Voyce polynomials via
Viennot's combinatorial theory of orthogonal polynomials.Comment: Minor correction
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