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## A FACTORIZATION THEOREM FOR LOZENGE TILINGS OF A HEXAGON WITH TRIANGULAR HOLES

### Citations

223 |
Combinatory Analysis,
- MacMahon
- 1915
(Show Context)
Citation Context ...o holes can be regarded as a new proof of the enumeration of symmetric plane partitions (first proved by Andrews [1]), as it follows, via our factorization result, from the base case (due to MacMahon =-=[17]-=-) and the transpose-complementary case (due to Proctor [19]), as we explain in Section 6. Our results are described in the next section. There are several other simple equations relating the symmetry ... |

210 |
Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture,
- Bressoud
- 1999
(Show Context)
Citation Context ...f plane partitions that fit in a given box — equivalently, symmetry classes of lozenge tilings of hexagons on the triangular lattice — forms a classical chapter of enumerative combinatorics (see [20],=-=[3]-=-,[2],[14],[22],[12]). Explicit product formulas exist for all symmetry classes, which makes it possible to find relations between them. One such striking relation is that M(H2a,b,b) = M−(H2a,b,b)M|(H2... |

157 | Nonintersecting paths, pfaffians, and plane partitions.
- Stembridge
- 1990
(Show Context)
Citation Context ...s the factor of 2n−2l on the right-hand side of (2.6), and the above claim follows. By the classical Lindström–Gessel–Viennot formula for the enumeration of nonintersecting lattice paths (see [16][9]=-=[21]-=-), the above weighted count can be written in terms of a determinant. Below we recall this formula. 10 M. CIUCU AND C. KRATTENTHALER Let G = (V,E) be a weighted directed acyclic graph with vertices V ... |

154 |
On the vector representations of induced matroids.
- Lindstrom
- 1973
(Show Context)
Citation Context ... cancels the factor of 2n−2l on the right-hand side of (2.6), and the above claim follows. By the classical Lindström–Gessel–Viennot formula for the enumeration of nonintersecting lattice paths (see =-=[16]-=-[9][21]), the above weighted count can be written in terms of a determinant. Below we recall this formula. 10 M. CIUCU AND C. KRATTENTHALER Let G = (V,E) be a weighted directed acyclic graph with vert... |

59 | Enumeration of perfect matchings in graphs with reflective symmetry, preprint
- Ciucu
- 1999
(Show Context)
Citation Context ...ical condensation (cf. [13]), while the transpose-complementary case (first proved by Proctor [19]) can be directly deduced from MacMahon’s result by applying the matchings factorization theorem (cf. =-=[4]-=-). From the point of view of [6], this adds a seventh symmetry class that can be proved in a combinatorial way. It would be interesting to find some more direct relations between the symmetry classes ... |

49 | Applications of graphical condensation for enumerating matchings and tilings
- Kuo
- 2004
(Show Context)
Citation Context ...t in the box and the number of those that are transpose-complementary. The formula for the total number (due to MacMahon [17]) can easily be proved inductively using Kuo’s graphical condensation (cf. =-=[13]-=-), while the transpose-complementary case (first proved by Proctor [19]) can be directly deduced from MacMahon’s result by applying the matchings factorization theorem (cf. [4]). From the point of vie... |

46 | Symmetries of plane partitions
- Stanley
- 1986
(Show Context)
Citation Context ...ses of plane partitions that fit in a given box — equivalently, symmetry classes of lozenge tilings of hexagons on the triangular lattice — forms a classical chapter of enumerative combinatorics (see =-=[20]-=-,[3],[2],[14],[22],[12]). Explicit product formulas exist for all symmetry classes, which makes it possible to find relations between them. One such striking relation is that M(H2a,b,b) = M−(H2a,b,b)M... |

44 |
Odd symplectic groups
- Proctor
- 1988
(Show Context)
Citation Context ...f symmetric plane partitions (first proved by Andrews [1]), as it follows, via our factorization result, from the base case (due to MacMahon [17]) and the transpose-complementary case (due to Proctor =-=[19]-=-), as we explain in Section 6. Our results are described in the next section. There are several other simple equations relating the symmetry classes of plane partitions which can be proved directly (s... |

37 |
The problem of the calissons
- David, Tomei
- 1989
(Show Context)
Citation Context ...an and the right-hand side in terms of a determinant. We start with the left-hand side. If we apply the standard translation of lozenge tilings to families of non-intersecting lattice paths (see e.g. =-=[8]-=- and [5]), then we obtain that Mf (Fn,2m(k1, k2, . . . , kl)) is equal to the number of all families (P−m+1, P−m+2, . . . , Pm, P1− , P2− , . . . , Pl− , P1+ , P2+ , . . . , Pl+) of non-intersecting l... |

35 |
Plane partitions V: the TSSCPP conjecture
- Andrews
- 1994
(Show Context)
Citation Context ...ane partitions that fit in a given box — equivalently, symmetry classes of lozenge tilings of hexagons on the triangular lattice — forms a classical chapter of enumerative combinatorics (see [20],[3],=-=[2]-=-,[14],[22],[12]). Explicit product formulas exist for all symmetry classes, which makes it possible to find relations between them. One such striking relation is that M(H2a,b,b) = M−(H2a,b,b)M|(H2a,b,... |

28 |
Determinants, paths and plane partitions, preprint
- Gessel, Viennot
- 1989
(Show Context)
Citation Context ...cels the factor of 2n−2l on the right-hand side of (2.6), and the above claim follows. By the classical Lindström–Gessel–Viennot formula for the enumeration of nonintersecting lattice paths (see [16]=-=[9]-=-[21]), the above weighted count can be written in terms of a determinant. Below we recall this formula. 10 M. CIUCU AND C. KRATTENTHALER Let G = (V,E) be a weighted directed acyclic graph with vertice... |

28 | On the generating functions for certain classes of plane partitions - Okada - 1989 |

22 | Notes on plane partitions - Gordon, Houten - 1968 |

22 |
An exploration of the permanent-determinant method, Electron
- Kuperberg
- 1998
(Show Context)
Citation Context ...in in Section 6. Our results are described in the next section. There are several other simple equations relating the symmetry classes of plane partitions which can be proved directly (see e.g. [5][6]=-=[15]-=-). There is still no unified proof available for all ten symmetry classes, but new direct ways of relating them to one another may help achieving this goal. 2. The factorization theorem Let n,m, l be ... |

7 |
Four symmetry classes of plane partitions under one roof
- Kuperberg
(Show Context)
Citation Context ...partitions that fit in a given box — equivalently, symmetry classes of lozenge tilings of hexagons on the triangular lattice — forms a classical chapter of enumerative combinatorics (see [20],[3],[2],=-=[14]-=-,[22],[12]). Explicit product formulas exist for all symmetry classes, which makes it possible to find relations between them. One such striking relation is that M(H2a,b,b) = M−(H2a,b,b)M|(H2a,b,b), (... |

6 | The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions
- Ciucu
- 1999
(Show Context)
Citation Context ... explain in Section 6. Our results are described in the next section. There are several other simple equations relating the symmetry classes of plane partitions which can be proved directly (see e.g. =-=[5]-=-[6][15]). There is still no unified proof available for all ten symmetry classes, but new direct ways of relating them to one another may help achieving this goal. 2. The factorization theorem Let n,m... |

3 |
Plane partitions I: The MacMahon conjecture. Studies in foundations and combinatorics
- Andrews
- 1978
(Show Context)
Citation Context ...ly placed holes along their horizontal symmetry axis. The special case when there are no holes can be regarded as a new proof of the enumeration of symmetric plane partitions (first proved by Andrews =-=[1]-=-), as it follows, via our factorization result, from the base case (due to MacMahon [17]) and the transpose-complementary case (due to Proctor [19]), as we explain in Section 6. Our results are descri... |

3 |
Plane partitions II: 5 1 symmetry classes
- Ciucu, Krattenthaler
- 2000
(Show Context)
Citation Context ...plain in Section 6. Our results are described in the next section. There are several other simple equations relating the symmetry classes of plane partitions which can be proved directly (see e.g. [5]=-=[6]-=-[15]). There is still no unified proof available for all ten symmetry classes, but new direct ways of relating them to one another may help achieving this goal. 2. The factorization theorem Let n,m, l... |

3 | A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings
- Ciucu, Krattenthaler
- 2010
(Show Context)
Citation Context ...ural questions: How can one see directly (without explicitly evaluating the terms) that the equation holds? And how can one generalize it? We presented a generalization in terms of Schur functions in =-=[7]-=-, which gave an algebraic reason for why equation (1.1) holds. In this paper we present a combinatorial generalization, in terms of hexagons that are allowed to have certain symmetrically placed holes... |

2 |
A proof of George Andrews
- Koutschan, Kauers, et al.
(Show Context)
Citation Context ... that fit in a given box — equivalently, symmetry classes of lozenge tilings of hexagons on the triangular lattice — forms a classical chapter of enumerative combinatorics (see [20],[3],[2],[14],[22],=-=[12]-=-). Explicit product formulas exist for all symmetry classes, which makes it possible to find relations between them. One such striking relation is that M(H2a,b,b) = M−(H2a,b,b)M|(H2a,b,b), (1.1) †Rese... |

2 |
The enumeration of totally symmetric plane partitions, Adv
- Stembridge
- 1995
(Show Context)
Citation Context ...tions that fit in a given box — equivalently, symmetry classes of lozenge tilings of hexagons on the triangular lattice — forms a classical chapter of enumerative combinatorics (see [20],[3],[2],[14],=-=[22]-=-,[12]). Explicit product formulas exist for all symmetry classes, which makes it possible to find relations between them. One such striking relation is that M(H2a,b,b) = M−(H2a,b,b)M|(H2a,b,b), (1.1) ... |

1 | Enumeration of symmetric centered rhombus tilings of a hexagon, preprint (2013), available at arxiv.org/abs/1306.1403
- Kasraoui, Krattenthaler
(Show Context)
Citation Context ...hey agree for all x. This completes the proof. A FACTORIZATION THEOREM FOR LOZENGE TILINGS 19 Remark. The case l = 0, x = 1 was first proved, by a different method, by Kasraoui and Krattenthaler in =-=[11]-=-. 6. Concluding remarks The factorization result proved in this paper is reminiscent of the symmetries considered by Kuperberg in [15, Sec. IV.C], especially given that, in the terminology of [15], on... |