5 research outputs found
Locked Polyomino Tilings
A locked -omino tiling is a grid tiling by -ominoes such that, if you
remove any pair of tiles, the only way to fill in the remaining grid cells
with -ominoes is to use the same two tiles in the exact same configuration
as before. We exclude degenerate cases where there is only one tiling overall
due to small dimensions. It is a classic (and straightforward) result that
finite grids do not admit locked 2-omino tilings. In this paper, we construct
explicit locked -omino tilings for on grids of various
dimensions. Most notably, we show that locked 3- and 4-omino tilings exist on
finite square grids of arbitrarily large size, and locked -omino tilings of
the infinite grid exist for arbitrarily large . The result for 4-omino
tilings in particular is remarkable because they are so rare and difficult to
construct: Only a single tiling is known to exist on any grid up to size .
Locked -omino tilings arise as obstructions to widely used political
redistricting algorithms in a model of redistricting where the underlying
census geography is a grid graph. Most prominent is the ReCom Markov chain,
which takes a random walk on the space of redistricting plans by iteratively
merging and splitting pairs of districts (tiles) at a time. Locked -omino
tilings are isolated states in the state space of ReCom. The constructions in
this paper are counterexamples to the meta-conjecture that ReCom is irreducible
on graphs of practical interest
Another involution principle-free bijective proof of Stanley's hook-content formula
Another bijective proof of Stanley's hook-content formula for the generating
function for semistandard tableaux of a given shape is given that does not
involve the involution principle of Garsia and Milne. It is the result of a
merge of the modified jeu de taquin idea from the author's previous bijective
proof (``An involution principle-free bijective proof of Stanley's hook-content
formula", Discrete Math. Theoret. Computer Science, to appear) and the
Novelli-Pak-Stoyanovskii bijection (Discrete Math. Theoret. Computer Science 1
(1997), 53-67) for the hook formula for standard Young tableaux of a given
shape. This new algorithm can also be used as an algorithm for the random
generation of tableaux of a given shape with bounded entries. An appropriate
deformation of this algorithm gives an algorithm for the random generation of
plane partitions inside a given box.Comment: 23 pages, AmS-Te
How to Couple from the Past Using a Read-Once Source of Randomness
We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure
LOZENGE TILING CONSTRAINED CODES
While the field of one-dimensional constrained codesis mature, with theoretical as well as practical aspects of codeanddecoder-design being well-established, such a theoreticaltreatment of its two-dimensional (2D) counterpart is still unavailable.Research has been conducted on a few exemplar2D constraints, e.g., the hard triangle model, run-length limitedconstraints on the square lattice, and 2D checkerboardconstraints. Excluding these results, 2D constrained systemsremain largely uncharacterized mathematically, with only loosebounds of capacities present. In this paper we present a lozengeconstraint on a regular triangular lattice and derive Shannonnoiseless capacity bounds. To estimate capacity of lozenge tilingwe make use of the bijection between the counting of lozengetiling and the counting of boxed plane partitions
Markov Chain Algorithms for Planar Lattice Structures (Extended Abstract)
Consider the following Markov chain, whose states are all domino tilings of a 2n x 2n chessboard: starting from some arbitrary tiling, pick a 2 x 2 window uniformly at random. If the four squares appearing in this window are covered by two parallel dominoes, rotate the dominoes in place. Repeat many times. This process is used in practice to generate a random tiling, and is a key tool in the study of the combinatorics of tilings and the behavior of dimer systems in statistical physics. Analogous Markov chains are used to randomly generate other structures on various twodimensional lattices. This paper presents techniques which prove for the first time that, in many interesting cases, a small number of random moves suffice to obtain a uniform distribution