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

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 $q$ and the $\tau$-enumeration of Plane Partitions with specific symmetries, with $\tau=-(q+q^{-1})$. We also find a conjectural relation \`a la Razumov-Stroganov between the $\tau\to 0$ 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

New Symmetric Plane Partition Identities from Invariant Theory Work of De Concini and Procesi

Nine (= 2 x 2 x 2 + 1) product identities for certain one-variable generating functions of certain families of plane partitions are presented in a unified fashion. The first two of these identities are originally due to MacMahon, Bender, Knuth, Gordon and Andrews and concern symmetric plane partitions. All nine identities are derived from tableaux descriptions of weights of especially nice representations of Lie groups, eight of them for the ‘right end node’ representations of SO˜(2n+1) and Sp(2n). The two newest identities come from a tableaux description which originally arose in work of De Concini and Procesi on classical invariant theory. All of the identities are of the most interest when viewed, in the context of plane partitions with symmetries contained in three-dimensional boxes

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

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