Generating functions for shifted plane partitions. (English) Zbl 0777.05009
Author’s introduction: Generating functions for certain families of plane partitions with a given shape and given row bounds can be written down in determinantal form. This has been shown by various methods (Chu, 1989; Gessel and Viennot, 1989; Krattenthaler, 1990; Wachs, 1985) the most prominent being the Gessel-Viennot (1985, 1989) method of nonintersecting lattice paths. In special cases these determinants can be evaluated to result into closed forms (Gessel and Viennot, 1989; Krattenthaler, 1990). Among them are the famous hook-content formulas of Stanley (1971).
With the help of a tableaux method, which is related to the nonintersecting path method, and a simple determinant evaluation, the author (Krattenthaler, 1990) provided a uniform proof method to establish essentially all known results concerning generating functions for plane partitions of a given shape (without symmetries) and, moreover, some new results. In this paper we continue this work. We transfer the method of Krattenthaler (1990) to the case of shifted plane partitions. These arise naturally when considering generating function problems for plane partitions with symmetries (Andrews, 1979; Mills et al., 1982, 1983, 1986; Okada, 1989). Using our tableaux method we derive determinant formulas for generating functions for shifted plane partitions of a given shape with prescribed entries on the main diagonal (Theorems 1 and 2 below).
Again there are special cases in which the determinants may be evaluated. With the help of a summation formula for Schur functions we are able to generalize Gansner’s (1981) hook formulas for trace generating functions for two particular families of shifted plane partitions.
A particular specialization of trace generating functions leads to the alternating trace generating functions of Proctor (1990). In the case of trapezoidal shapes, there are closed forms for the alternating trace generating function of shifted plane partitions with prescribed entries on the main diagonal. These formulas were first proved by Proctor (private communication) using group representation theory and bideterminants. With the help of a simple determinant evaluation (Lemma 7) we are able to provide alternative proofs which start with the determinantal formula of Theorem 1.
With the help of a tableaux method, which is related to the nonintersecting path method, and a simple determinant evaluation, the author (Krattenthaler, 1990) provided a uniform proof method to establish essentially all known results concerning generating functions for plane partitions of a given shape (without symmetries) and, moreover, some new results. In this paper we continue this work. We transfer the method of Krattenthaler (1990) to the case of shifted plane partitions. These arise naturally when considering generating function problems for plane partitions with symmetries (Andrews, 1979; Mills et al., 1982, 1983, 1986; Okada, 1989). Using our tableaux method we derive determinant formulas for generating functions for shifted plane partitions of a given shape with prescribed entries on the main diagonal (Theorems 1 and 2 below).
Again there are special cases in which the determinants may be evaluated. With the help of a summation formula for Schur functions we are able to generalize Gansner’s (1981) hook formulas for trace generating functions for two particular families of shifted plane partitions.
A particular specialization of trace generating functions leads to the alternating trace generating functions of Proctor (1990). In the case of trapezoidal shapes, there are closed forms for the alternating trace generating function of shifted plane partitions with prescribed entries on the main diagonal. These formulas were first proved by Proctor (private communication) using group representation theory and bideterminants. With the help of a simple determinant evaluation (Lemma 7) we are able to provide alternative proofs which start with the determinantal formula of Theorem 1.
Reviewer: J.Cigler (Wien)
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A18 | Partitions of sets |
| 05E05 | Symmetric functions and generalizations |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 11P81 | Elementary theory of partitions |
| 05E10 | Combinatorial aspects of representation theory |
Keywords:
plane partitions; lattice paths; tableaux; generating functions; determinants; Schur functions; hook formulasReferences:
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