Symmetries of plane partitions. (English) Zbl 0602.05007
J. Comb. Theory, Ser. A 43, 103-113 (1986); erratum ibid. 44, 310 (1987).
Author’s summary: ”We introduce a new symmetry operation, called complementation, on plane partitions whose three-dimensional diagram is contained in a given box. This operation was suggested by work of W. H. Mills, D. P. Robbins ad H. Rumsey jun. [J. Comb. Theory, Ser. A 34, 340–359 (1983; Zbl 0516.05016)]. There then arise a total of ten inequivalent problems concerned with the enumeration of plane partitions with a given symmetry. Four of these problems had been previously considered. We survey what is known about the problems and give a solution to one of them. The proof is based on the theory of Schur functions, in particular the Littlewood-Richardson rule. Of the ten problems, seven are now solved while the remaining three have conjectured simple solutions.”
[The erratum contains a corrected version of Theorem 3.4 after F. Brenti has pointed out that parts (b) and (c) of that Theorem are stated incorrectly in the paper (p. 111).]
[The erratum contains a corrected version of Theorem 3.4 after F. Brenti has pointed out that parts (b) and (c) of that Theorem are stated incorrectly in the paper (p. 111).]
Reviewer: Ian Anderson (Glasgow)
MSC:
| 05A17 | Combinatorial aspects of partitions of integers |
| 05A15 | Exact enumeration problems, generating functions |
Citations:
Zbl 0516.05016Online Encyclopedia of Integer Sequences:
Number of transpose-complementary plane partitions of n.Number of self-complementary plane partitions in a (2n)-cube.
References:
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