8 research outputs found
The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions
We prove that the number of cyclically symmetric, self-complementary plane
partitions contained in a cube of side equals the square of the number of
totally symmetric, self-complementary plane partitions contained in the same
cube, without explicitly evaluating either of these numbers. This appears to be
the first direct proof of this fact. The problem of finding such a proof was
suggested by Stanley
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
On the link pattern distribution of quarter-turn symmetric FPL configurations
We present new conjectures on the distribution of link patterns for
fully-packed loop (FPL) configurations that are invariant, or almost invariant,
under a quarter turn rotation, extending previous conjectures of Razumov and
Stroganov and of de Gier. We prove a special case, showing that the link
pattern that is conjectured to be the rarest does have the prescribed
probability. As a byproduct, we get a formula for the enumeration of a new
class of quasi-symmetry of plane partitions.Comment: 12 pages, 6 figures. Submitted to FPSAC 200
Half-turn symmetric FPLs with rare couplings and tilings of hexagons
14 p.International audienceIn this work, we put to light a formula that relies the number of fully packed loop configurations (FPLs) associated to a given coupling pi to the number of half-turn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling pi. When the coupling pi is the coupling with all arches parallel pi0 (the ''rarest'' one), this formula states the equality of the number of corresponding HTFPLs to the number of cyclically-symmetric plane partition of the same size. We provide a bijective proof of this fact. In the case of HTFPLs odd size, and although there is no similar expression, we study the number of HTFPLs whose coupling is a slit version of pi_0, and put to light new puzzling enumerative coincidence involving countings of tilings of hexagons and various symmetry classes of FPLs