106 research outputs found
Symmetries of plane partitions and the permanent-determinant method
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives
formulas for the number of plane partitions in each of ten symmetry classes.
This paper together with results by Andrews [J. Combin. Theory Ser. A 66
(1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the
project of proving all ten formulas.
We enumerate cyclically symmetric, self-complementary plane partitions. We
first convert plane partitions to tilings of a hexagon in the plane by
rhombuses, or equivalently to matchings in a certain planar graph. We can then
use the permanent-determinant method or a variant, the Hafnian-Pfaffian method,
to obtain the answer as the determinant or Pfaffian of a matrix in each of the
ten cases. We row-reduce the resulting matrix in the case under consideration
to prove the formula. A similar row-reduction process can be carried out in
many of the other cases, and we analyze three other symmetry classes of plane
partitions for comparison
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 Schur function identity related to the (-1)-enumeration of self-complementary plane partitions
We give another proof for the (-1)-enumeration of self-complementary plane
partitions with at least one odd side-length by specializing a certain Schur
function identity. The proof is analogous to Stanley's proof for the ordinary
enumeration. In addition, we obtain enumerations of 180-degree symmetric
rhombus tilings of hexagons with a barrier of arbitrary length along the
central line.Comment: AMSLatex, 14 pages, Parity conditions in Theorem 3 corrected and an
additional case adde
Four symmetry classes of plane partitions under one roof
In previous paper, the author applied the permanent-determinant method of
Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to
obtain a determinant or a Pfaffian that enumerates each of the ten symmetry
classes of plane partitions. After a cosmetic generalization of the Kasteleyn
method, we identify the matrices in the four determinantal cases (plain plane
partitions, cyclically symmetric plane partitions, transpose-complement plane
partitions, and the intersection of the last two types) in the representation
theory of sl(2,C). The result is a unified proof of the four enumerations
Enumeration of lozenge tilings of hexagons with cut off corners
Motivated by the enumeration of a class of plane partitions studied by
Proctor and by considerations about symmetry classes of plane partitions, we
consider the problem of enumerating lozenge tilings of a hexagon with ``maximal
staircases'' removed from some of its vertices. The case of one vertex
corresponds to Proctor's problem. For two vertices there are several cases to
consider, and most of them lead to nice enumeration formulas. For three or more
vertices there do not seem to exist nice product formulas in general, but in
one special situation a lot of factorization occurs, and we pose the problem of
finding a formula for the number of tilings in this case.Comment: 23 pages, AmS-Te
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