## Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole (0)

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Citations: | 23 - 9 self |

### BibTeX

@MISC{Ciucu_enumerationof,

author = {M. Ciucu and T. Eisenkölbl and T. Eisenk Olbl and D. Zare and C. Krattenthaler},

title = {Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole},

year = {}

}

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### Abstract

. 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...

### Citations

919 | Representation Theory - Fulton, Harris - 1991 |

683 | The Theory of Partitions - Andrews - 1976 |

661 | Basic Hypergeometric Series - Gasper, Rahman - 1990 |

434 | Symmetric functions and Hall polynomials, second edition - Macdonald - 1995 |

347 | Generalized Hypergeometric Functions - Slater - 1966 |

180 | Generalized Hypergeometric Series - Bailey - 1935 |

174 | Combinatory Analysis - MacMahon - 1916 |

116 |
On the vector representations of induced matroids
- Lindström
- 1973
(Show Context)
Citation Context ...A2, . . ., An, E1, E2, . . .,En be points of the planar integer lattice. Then the following identity holds: det 1≤i,j≤n (P(Ai → Ej)) = � (sgn σ) · P(A → Eσ, nonint.). (5.3) σ∈Sn Remark. The result in =-=[23]-=-, respectively [13], is in fact more general, as it is formulated for paths in an arbitrary oriented graph. But then the graph must satisfy an acyclicity ◦s20 M. CIUCU, T. EISENKÖLBL, C. KRATTENTHALER... |

75 | Advanced determinant calculus, Séminaire Lotharingien Combin - Krattenthaler |

39 | Symmetries of plane partitions
- Stanley
- 1986
(Show Context)
Citation Context ... there exists no core, we obtain enumeration results for cyclically symmetric plane partitions. Before we state these, let us briefly recall the relevant notions from plane partition theory (cf. e.g. =-=[36]-=- or [38, Sec. 1]). There are (at least) three possible equivalent ways to define plane partitions. Out of the three possibilities, in this paper, we choose to define a plane partition π as a subset of... |

33 | Plane partitions III: The weak Macdonald conjecture - Andrews, Andrews - 1979 |

30 |
The problem of the calissons
- David, Tomei
- 1989
(Show Context)
Citation Context ...ith side lengths 1 and angles of 60 ◦ and 120 ◦ . 1s2 M. CIUCU, T. EISENKÖLBL, C. KRATTENTHALER AND D. ZARE where H(n) stands for the “hyperfactorial” � n−1 k=0 k!. This follows from a bijection (cf. =-=[7]-=-) between such lozenge tilings and plane partitions contained in an a ×b×c box, and from MacMahon’s enumeration [25, Sec. 429, q → 1; proof in Sec. 494] of the latter. In [32] (see also [33]), Propp p... |

29 | Generating functions for plane partitions of a given shape - Krattenthaler - 1990 |

28 | Strange evaluations of hypergeometric series - Gessel, Stanton - 1982 |

27 | Multidimensional matrix inversions and Ar and Dr basic hypergeometric series
- Schlosser
- 1997
(Show Context)
Citation Context ...tensive theory of summation and transformation formulas (such a series is called a hypergeometric series in U(a) or an Aa hypergeometric series), mainly thanks to Milne and Gustafson (see for example =-=[14, 28, 29, 30, 34]-=-, and the references contained therein), it is only occasionally that series containing the square � 1≤i<j≤a (ki − kj) 2sLOZENGE TILINGS OF HEXAGONS WITH A CENTRAL TRIANGULAR HOLE 53 appear. Most of t... |

24 |
The Macdonald identities for affine root systems of classical type and hypergeometric series very well-poised on semi-simple Lie algebras
- Gustafson
- 1987
(Show Context)
Citation Context ...tensive theory of summation and transformation formulas (such a series is called a hypergeometric series in U(a) or an Aa hypergeometric series), mainly thanks to Milne and Gustafson (see for example =-=[14, 28, 29, 30, 34]-=-, and the references contained therein), it is only occasionally that series containing the square � 1≤i<j≤a (ki − kj) 2sLOZENGE TILINGS OF HEXAGONS WITH A CENTRAL TRIANGULAR HOLE 53 appear. Most of t... |

24 |
Jr., Enumeration of a symmetry class of plane partitions
- Mills, Robbins, et al.
- 1987
(Show Context)
Citation Context ...(−1) (a+1)/2 . It is then a routine computation to verify that this gives the multiplicative constant as claimed in (8.15). 9. Proof of Theorem 11 For the proof of Theorem 11, we proceed similarly to =-=[27]-=-. We define determinants Zn(x, µ) by � �n−1 � �� �� � i + µ k j − k + µ − 1 Zn(x, µ) = det −δij + x 0≤i,j≤n−1 t t j − k k−t � . (9.1) t,k=0 The only difference to the definition of Zn(x, µ) in [27] is... |

24 | On minuscule representations, plane partitions and involutions in complex Lie groups
- Stembridge
- 1994
(Show Context)
Citation Context ...unts, (−1)-enumerations of plane partitions, i.e., enumerations where plane partitions are given a weight of 1 or −1, according to certain rules, have been found to possess remarkable properties (see =-=[38, 39]-=-). Motivated in part by a conjectured (−1)-enumeration on cyclically symmetric plane partitions due to Stembridge [40], in Section 2 we consider a (−1)-enumeration of the lozenge tilings of Theorems 1... |

22 |
Determinants, paths, and plane partitions. Preprint, available at http://people.brandeis.edu/ gessel/homepage/papers/pp.pdf
- Gessel, Viennot
- 1989
(Show Context)
Citation Context ...E2, . . .,En be points of the planar integer lattice. Then the following identity holds: det 1≤i,j≤n (P(Ai → Ej)) = � (sgn σ) · P(A → Eσ, nonint.). (5.3) σ∈Sn Remark. The result in [23], respectively =-=[13]-=-, is in fact more general, as it is formulated for paths in an arbitrary oriented graph. But then the graph must satisfy an acyclicity ◦s20 M. CIUCU, T. EISENKÖLBL, C. KRATTENTHALER AND D. ZARE condit... |

22 | Perfect matchings and perfect squares - Jockusch - 1994 |

21 | An indefinite bibasic summation formula and some quadratic, cubic and quartic summation and transformation formulas, Canad - Gasper, Rahman - 1990 |

20 | Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions, Elect
- Krattenthaler
- 1997
(Show Context)
Citation Context ... (i + j + m/2)! × det (3j + m/2 + 1) . 0≤i,j≤a/2−1 (2i − j + m/2 + 1)! (2j − i)! = det 0≤i,j≤a/2−1 Both determinants on the right-hand side of this identity can be evaluated by means of Theorem 10 in =-=[17]-=-, which reads � � (x + y + i + j − 1)! det 0≤i,j≤n−1 (x + 2i − j)! (y + 2j − i)! = n−1 � i=0 This completes the proof of the theorem. i! (x + y + i − 1)! (2x + y + 2i)i (x + 2y + 2i)i . (9.6) (x + 2i)... |

20 |
Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics
- Propp
- 1999
(Show Context)
Citation Context ...ction (cf. [7]) between such lozenge tilings and plane partitions contained in an a ×b×c box, and from MacMahon’s enumeration [25, Sec. 429, q → 1; proof in Sec. 494] of the latter. In [32] (see also =-=[33]-=-), Propp posed several problems regarding “incomplete” hexagons. For example, Problem 2 in [32] (and [33]) asks for the number of lozenge tilings of a hexagon with side lengths n, n+1, n, n+1, n, n+1 ... |

17 | An alternative evaluation of the Andrews–Burge determinant
- Krattenthaler
- 1999
(Show Context)
Citation Context ...binations are themselves linearly independent. (Equivalently, we find m linearly independent vectors in the kernel of the linear operator defined by the matrix underlying D1(−e, c).) See Section 2 of =-=[18]-=-, and in particular the Lemma in that section, for a formal justification of this procedure.s24 M. CIUCU, T. EISENKÖLBL, C. KRATTENTHALER AND D. ZARE To be precise, we claim that the following equatio... |

16 |
An exploration of the permanent-determinant method, Electron
- Kuperberg
- 1998
(Show Context)
Citation Context ... the black lozenge has weight −1, all other lozenges have weight 1.) Yet another way to obtain this weight is through the perfect matchings point of view of lozenge tilings, elaborated for example in =-=[21, 22]-=-. In this setup, the cyclically symmetric lozenge tilings that we consider heres12 M. CIUCU, T. EISENKÖLBL, C. KRATTENTHALER AND D. ZARE correspond bijectively to perfect matchings in a certain hexago... |

16 | Some hidden relations involving the ten symmetry classes of plane partitions
- Stembridge
- 1994
(Show Context)
Citation Context ...unts, (−1)-enumerations of plane partitions, i.e., enumerations where plane partitions are given a weight of 1 or −1, according to certain rules, have been found to possess remarkable properties (see =-=[38, 39]-=-). Motivated in part by a conjectured (−1)-enumeration on cyclically symmetric plane partitions due to Stembridge [40], in Section 2 we consider a (−1)-enumeration of the lozenge tilings of Theorems 1... |

15 |
Twenty open problems on enumeration of matchings, manuscript
- Propp
- 1996
(Show Context)
Citation Context ...ows from a bijection (cf. [7]) between such lozenge tilings and plane partitions contained in an a ×b×c box, and from MacMahon’s enumeration [25, Sec. 429, q → 1; proof in Sec. 494] of the latter. In =-=[32]-=- (see also [33]), Propp posed several problems regarding “incomplete” hexagons. For example, Problem 2 in [32] (and [33]) asks for the number of lozenge tilings of a hexagon with side lengths n, n+1, ... |

14 |
Symmetries of plane partitions and the permanent determinant method
- Kuperberg
- 1994
(Show Context)
Citation Context ... the black lozenge has weight −1, all other lozenges have weight 1.) Yet another way to obtain this weight is through the perfect matchings point of view of lozenge tilings, elaborated for example in =-=[21, 22]-=-. In this setup, the cyclically symmetric lozenge tilings that we consider heres12 M. CIUCU, T. EISENKÖLBL, C. KRATTENTHALER AND D. ZARE correspond bijectively to perfect matchings in a certain hexago... |

14 |
very-well-poised 10φ9 transformations
- Milne, Newcomb, et al.
- 1996
(Show Context)
Citation Context ...tensive theory of summation and transformation formulas (such a series is called a hypergeometric series in U(a) or an Aa hypergeometric series), mainly thanks to Milne and Gustafson (see for example =-=[14, 28, 29, 30, 34]-=-, and the references contained therein), it is only occasionally that series containing the square � 1≤i<j≤a (ki − kj) 2sLOZENGE TILINGS OF HEXAGONS WITH A CENTRAL TRIANGULAR HOLE 53 appear. Most of t... |

14 | reprinted by Cambridge - Stanley - 1986 |

13 | Plane partitions I: a generalization of MacMahon’s formula, preprint (available at http://www.math.ias.edu/∼ciucu/genmac.ps
- Ciucu
(Show Context)
Citation Context ...d. This problem was solved in [4, Theorem 1], [15, Theorem 20] and [31, Theorem 1] (the most general result, for a hexagon with side lengths a, b + 1, c, a + 1, b, c + 1, being contained in [31]). In =-=[5]-=-, the first author considers the case when a larger triangle (in fact, possibly several) is removed. However, in contrast to [31], the results in [5] assume that the hexagon has a reflective symmetry,... |

9 | Enumeration of lozenge tilings of punctured hexagons - Ciucu - 1998 |

9 | Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus in the centre
- Fischer
(Show Context)
Citation Context ...= 1, 2, . . ., n. (The only exceptions that we are aware of, i.e., applications of the above formula in the case where the sum on the right-hand side does not reduce to a single term, can be found in =-=[8]-=-, [23], and [41].) This is, however, not exactly the situation that we encounter in our problem. Therefore, it seems that Lemma 14 is not suited for our problem. However, our choice of starting and en... |

9 | Function identities and the number of perfect matchings of holey Aztec rectangles, in: “q–series from a Contemporary Perspective - Krattenthaler, Schur - 2000 |

9 |
Ueber die Funktionen, welche durch Reichen von der Form dargestellst verden 1
- Thomae
(Show Context)
Citation Context ...a+m (1 + a + c − e + 2 m − s)−1+e−m+s (1 + a − e + 2 m − s)−1+e−m+s � � 1 − c − i − m, 1 − e + m − s, 1 − a − m × 3F2 ; 1 . 1 − a − c − m, 2 − e − i Next we use a transformation formula due to Thomae =-=[42]-=- (see also [10, (3.1.1)]), � � A, B, −n 3F2 ; 1 = D, E (E − B)n � � −n, B, D − A 3F2 ; 1 , (7.7) (E)n D, 1 + B − E − n where n is a nonnegative integer. This gives (1 + a + c − e + 2 m − s)e−m+s−1 (1 ... |

9 | Consequences of the A` and C` Bailey transform and Bailey lemma, Discrete Math - Milne, Lilly - 1995 |

8 |
A q-analog of a Whipple’s transformation for hypergeometric series
- Milne
- 1994
(Show Context)
Citation Context |

8 | The Macdonald identities for ane root systems of classical type and hypergeometric series very well-poised on semi-simple Lie algebras - Gustafson - 1987 |

7 | Tilings of diamonds and hexagons with defects”, Electron - Helfgott, Gessel - 1999 |

7 | The number of rhombus tilings of a “punctured” hexagon and the minor summation formula
- Okada, Krattenthaler
- 1998
(Show Context)
Citation Context ...gle removed. This problem was solved in [4, Theorem 1], [15, Theorem 20] and [31, Theorem 1] (the most general result, for a hexagon with side lengths a, b + 1, c, a + 1, b, c + 1, being contained in =-=[31]-=-). In [5], the first author considers the case when a larger triangle (in fact, possibly several) is removed. However, in contrast to [31], the results in [5] assume that the hexagon has a reflective ... |

7 | Multidimensional matrix inversion and A r and D r basic hypergeometric series - Schlosser - 1997 |

6 | The number of rhombus tilings of a "punctured" hexagon and the minor summation formula - Okada, Krattenthaler - 1998 |

5 |
A determinant for q-counting lattice paths
- Sulanke
- 1990
(Show Context)
Citation Context ...n. (The only exceptions that we are aware of, i.e., applications of the above formula in the case where the sum on the right-hand side does not reduce to a single term, can be found in [8], [23], and =-=[41]-=-.) This is, however, not exactly the situation that we encounter in our problem. Therefore, it seems that Lemma 14 is not suited for our problem. However, our choice of starting and end points (see Fi... |

4 | Exact enumeration of tilings of diamonds and hexagons with defects, preprint - Helfgott, Gessel |

3 | Plane partitions II: 5 1 symmetry classes, in: Combinatorial Methods in Representation Theory - Ciucu, Krattenthaler |

3 | U(n) very-well-poised 10 - Milne, Newcomb - 1996 |

3 | Enumeration of matchings: Problems and progress, in: "New Perspectives in Algebraic Combinatorics - Propp - 1999 |

2 | The number of rhombus tilings of a \punctured" hexagon and the minor summation formula - Okada, Krattenthaler - 1998 |

1 |
Strange enumerations of CSPP's and TSPP's, unpublished manuscript
- Stembridge
- 1993
(Show Context)
Citation Context ...rding to certain rules, have been found to possess remarkable properties (see [38, 39]). Motivated in part by a conjectured (−1)-enumeration on cyclically symmetric plane partitions due to Stembridge =-=[40]-=-, in Section 2 we consider a (−1)-enumeration of the lozenge tilings of Theorems 1 and 2. The corresponding results are given in Theorems 4 and 5. In Section 3, we restrict our attention to cyclically... |

1 | Plane partitions II: 5 2 symmetry classes, in: Combinatorial Methods in Representation Theory - Ciucu, Krattenthaler |