175 research outputs found
(-1)-enumeration of plane partitions with complementation symmetry
We compute the weighted enumeration of plane partitions contained in a given
box with complementation symmetry where adding one half of an orbit of cubes
and removing the other half of the orbit changes the weight by -1 as proposed
by Kuperberg. We use nonintersecting lattice path families to accomplish this
for transpose-complementary, cyclically symmetric transpose-complementary and
totally symmetric self-complementary plane partitions. For symmetric
transpose-complementary and self-complementary plane partitions we get partial
results. We also describe Kuperberg's proof for the case of cyclically
symmetric self-complementary plane partitions.Comment: 41 pages, AmS-LaTeX, uses TeXDraw; reference adde
Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries
We propose new conjectures relating sum rules for the polynomial solution of
the qKZ equation with open (reflecting) boundaries as a function of the quantum
parameter and the -enumeration of Plane Partitions with specific
symmetries, with . We also find a conjectural relation \`a la
Razumov-Stroganov between the limit of the qKZ solution and refined
numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision
Rhombus Tilings of a Hexagon with Three Fixed Border Tiles
We compute the number of rhombus tilings of a hexagon with sides
with three fixed tiles touching the border. The
particular case solves a problem posed by Propp. Our result can also be
viewed as the enumeration of plane partitions having rows and
columns, with largest entry , with a given number of entries in
the first row, a given number of entries 0 in the last column and a given
bottom-left entry.Comment: 7 pages, AmS-LaTeX, uses TeXDraw; revised version which is to appear
in J. Combin. Theory Ser.
On some ground state components of the O(1) loop model
We address a number of conjectures about the ground state O(1) loop model,
computing in particular two infinite series of partial sums of its entries and
relating them to the enumeration of plane partitions. Our main tool is the use
of integral formulae for a polynomial solution of the quantum
Knizhnik-Zamolodchikov equation.Comment: v4: fixed problem with refs appearing as
Double Aztec Rectangles
We investigate the connection between lozenge tilings and domino tilings by
introducing a new family of regions obtained by attaching two different Aztec
rectangles. We prove a simple product formula for the generating functions of
the tilings of the new regions, which involves the statistics as in the Aztec
diamond theorem (Elkies, Kuperberg, Larsen, and Propp, J. Algebraic Combin.
1992). Moreover, we consider the connection between the generating function and
MacMahon's -enumeration of plane partitions fitting in a given boxComment: 16 page
Monomial Complete Intersections, The Weak Lefschetz Property and Plane Partitions
We characterize the monomial complete intersections in three variables
satisfying the Weak Lefschetz Property (WLP), as a function of the
characteristic of the base field. Our result presents a surprising, and still
combinatorially obscure, connection with the enumeration of plane partitions.
It turns out that the rational primes p dividing the number, M(a,b,c), of plane
partitions contained inside an arbitrary box of given sides a,b,c are precisely
those for which a suitable monomial complete intersection (explicitly
constructed as a bijective function of a,b,c) fails to have the WLP in
characteristic p. We wonder how powerful can be this connection between
combinatorial commutative algebra and partition theory. We present a first
result in this direction, by deducing, using our algebraic techniques for the
WLP, some explicit information on the rational primes dividing M(a,b,c).Comment: 16 pages. Minor revisions, mainly to keep track of two interesting
developments following the original posting. Final version to appear in
Discrete Mat
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