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

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

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

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 $2n$ 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

(-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

Kasteleyn cokernels

We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or $q$-round, and we conjecture that cokernels remain round or $q$-round for related ``impossible enumerations'' in which there are no tilings. Our conjectures provide a new view of the topic of enumerating symmetry classes of plane partitions and their generalizations. In particular we conjecture that a $q$-specialization of a Jacobi-Trudi matrix has a Smith normal form. If so it could be an interesting structure associated to the corresponding irreducible representation of \SL(n,\C). Finally we find, with proof, the normal form of the matrix that appears in the enumeration of domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction