Generating functions for plane partitions of a given shape

For fixed integers α and β, planar arrays of integers of a given shape, in which the entries decrease at least by α along rows and at least by β along columns, are considered. For various classes of these (α,β)-plane partitions we compute three different kinds of generating functions. By a combinatorial method, determinantal expressions are obtained for these generating functions. In special cases these determinants may be evaluated by a simple determinant lemma. All known results concerning plane partitions of a given shape are included. Thus our approach of a given shape provides a uniform proof method and yields numerous generalizations of known results.

[1]  Leonard Carlitz,et al.  Rectangular arrays and plane partitions , 1967 .

[2]  A. P. Hillman,et al.  Reverse Plane Partitions and Tableau Hook Numbers , 1976, J. Comb. Theory A.

[3]  Robert A. Proctor.  Shifted plane partitions of trapezoidal shape , 1983 .

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

[5]  Donald E. Knuth,et al.  PERMUTATIONS, MATRICES, AND GENERALIZED YOUNG TABLEAUX , 1970 .

[6]  Emden R. Gansner,et al.  The Hillman-Grassl Correspondence and the Enumeration of Reverse Plane Partitions , 1981, J. Comb. Theory, Ser. A.

[7]  Michelle L. Wachs,et al.  Flagged Schur Functions, Schubert Polynomials, and Symmetrizing Operators , 1985, J. Comb. Theory, Ser. A.

[8]  R. Stanley.  Ordered Structures And Partitions , 1972 .

[9]  J. Remmel,et al.  A bijective proof of the generating function for the number of reverse plane partitions via lattice paths , 1984 .

[10]  D. White,et al.  Constructive combinatorics , 1986 .

[11]  Doron Zeilberger.  Andre's reflection proof generalized to the many-candidate ballot problem , 1983, Discret. Math..

[12]  Basil Gordon,et al.  A proof of the Bender-Knuth conjecture. , 1983 .

[13]  I. Gessel,et al.  Binomial Determinants, Paths, and Hook Length Formulae , 1985 .

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

[15]  W. Rheinboldt,et al.  Generalized hypergeometric functions , 1968 .

[16]  G. Carpenter.  In Providence, R. I. , 1929 .

[17]  Christian Krattenthaler,et al.  Enumeration of lattice paths and generating functions for skew plane partitions , 1989 .

[18]  Jeffrey B. Remmel,et al.  A Bijective Proof of the Hook Formula for the Number of Column Strict Tableaux with Bounded Entries , 1983, Eur. J. Comb..

[19]  E. Gansner.  The enumeration of plane partitions via the Burge correspondence , 1981 .

[20]  Robert A. Proctor.  Bruhat Lattices, Plane Partition Generating Functions, and Minuscule Representations , 1984, Eur. J. Comb..

[21]  C. Krattenthaler.  A determinant evaluation and some enumeration results for plane partitions , 1990 .

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

[23]  Edward A. Bender,et al.  Enumeration of Plane Partitions , 1972, J. Comb. Theory A.

[24]  R. Stanley.  Theory and applications of plane partitions: Part 1 , 1971 .

[25]  G. B. Mathews,et al.  Combinatory Analysis. Vol. II , 1915, The Mathematical Gazette.

[26]  E. Hille,et al.  Analysis, Vol. 2. , 1969 .

[27]  Richard P. Stanley,et al.  The Conjugate Trace and Trace of a Plane Partition , 1973, J. Comb. Theory, Ser. A.