14,840 research outputs found
A factorization theorem for lozenge tilings of a hexagon with triangular holes
In this paper we present a combinatorial generalization of the fact that the
number of plane partitions that fit in a box is equal to
the number of such plane partitions that are symmetric, times the number of
such plane partitions for which the transpose is the same as the complement. We
use the equivalent phrasing of this identity in terms of symmetry classes of
lozenge tilings of a hexagon on the triangular lattice. Our generalization
consists of allowing the hexagon have certain symmetrically placed holes along
its horizontal symmetry axis. The special case when there are no holes can be
viewed as a new, simpler proof of the enumeration of symmetric plane
partitions.Comment: 20 page
An affine generalization of evacuation
We establish the existence of an involution on tabloids that is analogous to
Schutzenberger's evacuation map on standard Young tableaux. We find that the
number of its fixed points is given by evaluating a certain Green's polynomial
at , and satisfies a "domino-like" recurrence relation.Comment: 32 pages, 7 figure
Truncated determinants and the refined enumeration of Alternating Sign Matrices and Descending Plane Partitions
Lecture notes for the proceedings of the workshop "Algebraic Combinatorics
related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S.,
Nara, Japan.Comment: 25 pages, 8 figure
Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation
with reflecting boundary conditions which is relevant to the Temperley--Lieb
model of loops on a strip. By use of integral formulae we prove conjectures
relating it to the weighted enumeration of Cyclically Symmetric Transpose
Complement Plane Partitions and related combinatorial objects
Reconceiving the Ninth Amendment
The courts long have protected constitutional rights that are not listed explicitly in the Constitution, but are they warranted in doing so? As scholars and commentators vigorously debate this and other questions about the appropriate role of judges in interpreting the Constitution, the Ninth Amendment has assumed increasing importance. Its declaration that [t]he enumeration in the Constitution, of certain rights, shall not be construed to deny or disparage others retained by the people has suggested to many that the set of rights protected by the Constitution is not dosed and that judges may be authorized to protect these unenumerated rights on occasion
Deformed Kazhdan-Lusztig elements and Macdonald polynomials
We introduce deformations of Kazhdan-Lusztig elements and specialised
nonsymmetric Macdonald polynomials, both of which form a distinguished basis of
the polynomial representation of a maximal parabolic subalgebra of the Hecke
algebra. We give explicit integral formula for these polynomials, and
explicitly describe the transition matrices between classes of polynomials. We
further develop a combinatorial interpretation of homogeneous evaluations using
an expansion in terms of Schubert polynomials in the deformation parameters.Comment: major revision, 29 pages, 22 eps figure
Cluster expansion for abstract polymer models. New bounds from an old approach
We revisit the classical approach to cluster expansions, based on tree
graphs, and establish a new convergence condition that improves those by
Kotecky-Preiss and Dobrushin, as we show in some examples. The two ingredients
of our approach are: (i) a careful consideration of the Penrose identity for
truncated functions, and (ii) the use of iterated transformations to bound
tree-graph expansions.Comment: 16 pages. This new version, written en reponse to the suggestions of
the referees, includes more detailed introductory sections, a proof of the
generalized Penrose identity and some additional results that follow from our
treatmen
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