Graphical condensation, overlapping Pfaffians and superpositions of matchings

The purpose of this note is to exhibit clearly how the "graphical condensation" identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn-Percus method for the enumeration of matchings. Knuth termed the relevant identities "overlapping Pfaffian" identities and the key concept of proof "superpositions of matchings". In our uniform presentation of the material, we also give an apparently unpublished general "overlapping Pfaffian" identity of Krattenthaler. A previous version of this paper contained an erroneous application of the Kasteleyn-Percus method, which is now corrected.Comment: Th error in the application of the Kasteleyn-Percus method has been correcte

Variations on a theme of Kasteleyn, with application to the totally nonnegative Grassmannian

We provide a short proof of a classical result of Kasteleyn, and prove several variants thereof. One of these results has become key in the parametrization of positroid varieties, and thus deserves the short direct proof which we provide

A one-parameter generalization of Pfaffians

In analogy to the definition of the lambda-determinant, we define a one-parameter deformation of the Dodgson condensation formula for Pfaffians. We prove that the resulting rational function is a polynomial with weights given by the crossings and nestings of perfect matchings and prove several identities and closed-form evaluations.Comment: 12 pages, 6 figure

Proof of a conjecture of Kenyon and Wilson on semicontiguous minors

Kenyon and Wilson showed how to test if a circular planar electrical network with $n$ nodes is well-connected by checking the positivity of $\binom{n}{2}$ central minors of the response matrix. Their test is based on the fact that any contiguous minor of a matrix can be expressed as a Laurent polynomial in the central minors. Moreover, the Laurent polynomial is the generating function of domino tilings of a weighted Aztec diamond. They conjectured that a larger family of minors, semicontiguous minors, can also be written in terms of domino tilings of a region on the square lattice. In this paper, we present a proof of the conjecture.Comment: Final version: 33 pages. To appear in Journal of Combinatorial Theory, Series

A generalization of Kuo condensation

Kuo introduced his 4-point condensation in 2003 for bipartite planar graphs. In 2006 Kuo generalized this 4-point condensation to planar graphs that are not necessarily bipartite. His formula expressed the product between the number of perfect matching of the original graph $G$ and that of the subgraph obtained from $G$ by removing the four distinguished vertices as a Pfaffian of order 4, whose entries are numbers of perfect matchings of subgraphs of $G$ obtained by removing various pairs of vertices chosen from among the four distinguished ones. The compelling elegance of this formula is inviting of generalization. Kuo generalized it to $2k$ points under the special assumption that the subgraph obtained by removing some subset of the $2k$ vertices has precisely one perfect matching. In this paper we prove that the formula holds in the general case. We also present a couple of applications.Comment: 22 page

A qq-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary

MacMahon proved a simple product formula for the generating function of plane partitions fitting in a given box. The theorem implies a $q$-enumeration of lozenge tilings of a semi-regular hexagon on the triangular lattice. In this paper we generalize MacMahon's classical theorem by $q$-enumerating lozenge tilings of a new family of hexagons with four adjacent triangles removed from their boundary.Comment: 30 pages. Title is changed from "A q-enumeration of generalized plane partitions" to "A q-enumeration of lozenge tilings of a hexagon with four adjacent triangles removed from the boundary

A qq-enumeration of lozenge tilings of a hexagon with three dents

We $q$-enumerate lozenge tilings of a hexagon with three bowtie-shaped regions have been removed from three non-consecutive sides. The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths $2n,2n+3,2n,2n+3,2n,2n+3$ (in cyclic order) with the central unit triangles on the $(2n+3)$-sides removed.Comment: 33 pages. The title has been change

Plane partitions in the work of Richard Stanley and his school

These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.Comment: AmS-LaTeX, 33 pages; article written for the 70th-birthday volume honoring Richard Stanley; some comments on the relationsship between non-intersecting lattice paths and Kasteleyn matrices adde

Lozenge tilings of hexagons with central holes and dents

Ciucu showed that the number of lozenge tilings of a hexagon in which a chain of equilateral triangles of alternating orientations, called a `\emph{fern}', has been removed in the center is given by a simple product formula (Adv. Math. 2017). In this paper, we present a multi-parameter generalization of this work by giving an explicit tiling enumeration for a hexagon with three ferns removed, besides the middle fern located in the center as in Ciucu's region, we remove two additional ferns from two sides of the hexagon. Our result also implies a counterpart of MacMahon's classical formula of boxed plane partitions, corresponding the \emph{exterior} of the union of three disjoint concave polygons obtained by turning 120 degrees after drawing each side.Comment: Version 6: 52 pages. Version 6 is roughly the first half of Version 5, which also contained the material that is now arXiv:1905.07119. Version 6 contains minor expository change

Beyond Aztec Castles: Toric Cascades in the dP3dP_3 Quiver

Given one of an infinite class of supersymmetric quiver gauge theories, string theorists can associate a corresponding toric variety (which is a Calabi-Yau 3-fold) as well as an associated combinatorial model known as a brane tiling. In combinatorial language, a brane tiling is a bipartite graph on a torus and its perfect matchings are of interest to both combinatorialists and physicists alike. A cluster algebra may also be associated to such quivers and in this paper we study the generators of this algebra, known as cluster variables, for the quiver associated to the cone over the del Pezzo surface $dP_3$. In particular, mutation sequences involving mutations exclusively at vertices with two in-coming arrows and two out-going arrows are referred to as toric cascades in the string theory literature. Such toric cascades give rise to interesting discrete integrable systems on the level of cluster variable dynamics. We provide an explicit algebraic formula for all cluster variables which are reachable by toric cascades as well as a combinatorial interpretation involving perfect matchings of subgraphs of the $dP_3$ brane tiling for these formulas in most cases.Comment: Typos on page 42 fixed, final version to appear in Comm. Math. Phy