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Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole
"... . We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)enume ..."
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. We deal with the unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths a; b + m; c; a + m; b; c + m, where an equilateral triangle of side length m has been removed from the center. We give closed formulas for the plain enumeration and for a certain (\Gamma1)enumeration of these lozenge tilings. In the case that a = b = c, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For m = 0, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn\Gamma1 \Gamma !ffi ij + \Gamma m+i+j j \Delta\Delta , w...
Enumeration of lozenge tilings of hexagons with cut off corners
 J. Comb. Th. Ser. A
"... Abstract. 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 ..."
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Abstract. 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.
An Exploration of the PermanentDeterminant Method
 Electron. J. Combin.5
, 1998
"... The permanentdeterminant method and its generalization, the HafnianPfa #an method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanentdeterminant with consequences in enumera ..."
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The permanentdeterminant method and its generalization, the HafnianPfa #an method, are methods to enumerate perfect matchings of plane graphs that were discovered by P. W. Kasteleyn. We present several new techniques and arguments related to the permanentdeterminant with consequences in enumerative combinatorics. Here are some of the results that follow from these techniques: 1. If a bipartite graph on the sphere with 4n vertices is invariant under the antipodal map, the number of matchings is the square of the number of matchings of the quotient graph. 2. The number of matchings of the edge graph of a graph with vertices of degree at most 3 is a power of 2. 3. The three Carlitz matrices whose determinants count a b c plane partitions all have the same cokernel. 4. Two symmetry classes of plane partitions can be enumerated with almost no calculation. Submitted: October 16, 1998; Accepted: November 9, 1998 [Also available as math.CO/9810091] The permanentdeterminan...
Plane partitions I: A generalization of MacMahon’s formula
 Memoirs Amer. Math. Soc
"... Abstract. The number of plane partitions contained in a given box was shown by MacMahon [8] to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of sidelengths a, b, c, a, b, c (in cyclic order) and angles of 120 degrees. We presen ..."
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Abstract. The number of plane partitions contained in a given box was shown by MacMahon [8] to be given by a simple product formula. By a simple bijection, this formula also enumerates lozenge tilings of hexagons of sidelengths a, b, c, a, b, c (in cyclic order) and angles of 120 degrees. We present a generalization in the case b = c by giving simple product formulas enumerating lozenge tilings of the regions obtained from a hexagon of sidelengths a, b + k, b, a + k, b, b + k (where k is an arbitrary nonnegative integer) and angles of 120 degrees by removing certain triangular regions along its symmetry axis. 1.
Tilings of diamonds and hexagons with defects”, Electron
 J. Combin
, 1999
"... Abstract. We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to open problems 1, 2, and 10 in James Propp’s list of problems on enumeration of matchings [ ..."
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Abstract. We show how to count tilings of Aztec diamonds and hexagons with defects using determinants. In several cases these determinants can be evaluated in closed form. In particular, we obtain solutions to open problems 1, 2, and 10 in James Propp’s list of problems on enumeration of matchings [21]. 1.
Enumeration of matchings: problems and progress
 in New Perspectives in Algebraic Combinatorics
, 1999
"... Abstract. This document is built around a list of thirtytwo problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary ..."
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Cited by 12 (0 self)
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Abstract. This document is built around a list of thirtytwo problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of matchings. The twenty original problems, with commentary, comprise the bulk of the article. I give an account of the progress that has been made on these problems as of this writing, and include pointers to both the printed and online literature; roughly half of the original twenty problems were solved by participants in the MSRI Workshop on Combinatorics, their students, and others, between 1996 and 1999. The article concludes with a dozen new open problems. 1.
The Number of Rhombus Tilings of a "Punctured" Hexagon and the Minor Summation Formula
, 1998
"... We compute the number of all rhombus tilings of a hexagon with sides a; b + 1; c; a+1; b; c+ 1, of which the central triangle is removed, provided a; b; c have the same parity. The result is B(d a 2 e; d b 2 e; d c 2 e)B(d a+1 2 e; b b 2 c; d c 2 e)B(d a 2 e; d b+1 2 e; b c ..."
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We compute the number of all rhombus tilings of a hexagon with sides a; b + 1; c; a+1; b; c+ 1, of which the central triangle is removed, provided a; b; c have the same parity. The result is B(d a 2 e; d b 2 e; d c 2 e)B(d a+1 2 e; b b 2 c; d c 2 e)B(d a 2 e; d b+1 2 e; b c 2 c)B(b a 2 c; d b 2 e; d c+1 2 e), where B(ff; fi; fl) is the number of plane partitions inside the ff \Theta fi \Theta fl box. The proof uses nonintersecting lattice paths and a new identity for Schur functions, which is proved by means of the minor summation formula of Ishikawa and Wakayama. A symmetric generalization of this identity is stated as a conjecture. 1.
Enumeration of symmetric centered rhombus tilings of a hexagon, preprint (2013), available at arxiv.org/abs/1306.1403
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HALFTURN SYMMETRIC FPLs WITH RARE COUPLINGS AND TILINGS OF HEXAGONS
"... Abstract. In this work, we put to light a formula that relies the number of fully packed loop configurations (FPLs) associated to a given coupling π to the number of halfturn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling π. When the coupling π is the cou ..."
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Abstract. In this work, we put to light a formula that relies the number of fully packed loop configurations (FPLs) associated to a given coupling π to the number of halfturn symmetric FPLs (HTFPLs) of even size whose coupling is a punctured version of the coupling π. When the coupling π is the coupling with all arches parallel π0 (the “rarest ” one), this formula states the equality of the number of corresponding HTFPLs to the number of cyclicallysymmetric 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 π0, and put to light new puzzling enumerative coincidence involving countings of tilings of hexagons and various symmetry classes of FPLs.