Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole

We deal with 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 (?1)-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 det0?i, j?n?1(??ij+(m+i+jj)), where ? is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (1979, Invent. Math.53, 193?225), which corresponds to ?=1.

[1]  Ira M. Gessel,et al.  Strange Evaluations of Hypergeometric Series , 1982 .

[2]  Richard P. Stanley,et al.  Symmetries of plane partitions , 1986, J. Comb. Theory A.

[3]  Twenty Open Problems in Enumeration of Matchings , 1998, math/9801060.

[4]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[5]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[6]  Mizan Rahman,et al.  An Indefinite Bibasic Summation Formula and Some Quadratic, Cubic and Quartic Summation and Transformation Formulas , 1990, Canadian Journal of Mathematics.

[7]  Gian-Carlo Rota,et al.  Mathematical Essays in honor of Gian-Carlo Rota , 1998 .

[8]  J. Thomae.  Ueber die Functionen, welche durch Reihen von der Form dargestellt werden . , 1879 .

[9]  Ravi P. Agarwal.  Generalized hypergeometric series , 1963 .

[10]  Christopher Parker Sonderforschungsbereich,et al.  Minuscule Representations , 1993 .

[11]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[12]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[14]  G. David,et al.  The problem of Calissons , 1989 .

[15]  John R. Stembridge,et al.  Some Hidden Relations Involving the Ten Symmetry Classes of Plane Partitions , 1994, J. Comb. Theory, Ser. A.

[16]  William Jockusch.  Perfect Matchings and Perfect Squares , 1994, J. Comb. Theory, Ser. A.

[17]  Christian Krattenthaler,et al.  An Alternative Evaluation of the Andrews-Burge Determinant , 1998 .

[18]  Stephen C. Milne,et al.  Consequences of the Al and Cl Bailey transform and Bailey lemma , 1995, Discret. Math..

[19]  Greg Kuperberg.  An Exploration of the Permanent-Determinant Method , 1998, Electron. J. Comb..

[20]  U ( n ) very-well-posed 10 p 9 transformations , 1996 .

[21]  Christian Krattenthaler,et al.  The Number of Rhombus Tilings of a “Punctured” Hexagon and the Minor Summation Formula , 1997 .

[22]  Mihai Ciucu,et al.  Plane partitions II: 5 1/2 symmetry classes , 1998 .

[23]  J. Shaw.  Combinatory Analysis , 1917, Nature.

[24]  G. Andrews.  The Theory of Partitions: Frontmatter , 1976 .

[25]  Mihai Ciucu.  Enumeration of Lozenge Tilings of Punctured Hexagons , 1998, J. Comb. Theory, Ser. A.

[26]  Ira M. Gessel,et al.  ENUMERATION OF TILINGS OF DIAMONDS AND HEXAGONS WITH DEFECTS , 1998 .

[27]  Christian Krattenthaler.  Determinant Identities and a Generalization of the Number of Totally Symmetric Self-complementary Plane Partitions , 1997, Electron. J. Comb..

[28]  Christian Krattenthaler,et al.  Generating functions for plane partitions of a given shape , 1990 .

[29]  John R. Stembridge.  On minuscule representations, plane partitions and involutions in complex Lie groups , 1994 .

[30]  George E. Andrews,et al.  Plane partitions (III): The weak Macdonald conjecture , 1979 .

[31]  Robert A. Sulanke,et al.  A determinant for q-counting n-dimensional lattice paths , 1990, Discret. Math..

[32]  Michael J. Schlosser,et al.  Multidimensional Matrix Inversions and Ar and Dr Basic Hypergeometric Series , 1997 .

[33]  Ilse Fischer.  Enumeration of Rhombus Tilings of a Hexagon which Contain a Fixed Rhombus in the Centre , 2001, J. Comb. Theory, Ser. A.

[34]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[35]  B. Dwork.  Generalized Hypergeometric Functions , 1990 .

[36]  Greg Kuperberg.  Symmetries of Plane Partitions and the Permanent - Determinant Method , 1994, J. Comb. Theory, Ser. A.

[37]  W. H. Mills,et al.  Proof of the Macdonald conjecture , 1982 .

[38]  C. Krattenthaler.  Schur function identities and the number of perfect matchings of holey Aztec rectangles , 1997, math/9712204.

[39]  B. Lindström.  On the Vector Representations of Induced Matroids , 1973 .

[40]  J. Propp.  Enumeration of Matchings: problems and Progress , 1999, math/9904150.

[41]  David P. Robbins,et al.  Enumeration of a symmetry class of plane partitions , 1987, Discret. Math..