Self-complementary totally symmetric plane partitions
type: Item
instance of: scholarly article
title: Self-complementary totally symmetric plane partitions
zbMATH Open document ID: 0615.05011
DOI: 10.1016/0097-3165(86)90098-1
author: W. H. Mills
author: David P. Robbins
author: Howard jun. Rumsey
published in: Journal of Combinatorial Theory. Series A
publication date: 1986-01-01T00:00:00Z
review text: A totally symmetric plane partition of size n is a plane partition whose three-dimensional Ferrers graph is contained in the box \(X_ n=[1,n]\times [1,n]\times [1,n]\) and which is mapped to itself under all permutations of the coordinate axes. The complement of the Ferrers graph of such a plane partition (that is, the set of lattice points in the box \(X_ n\) that do not belong to the Ferrers graph) is again totally symmetric when viewed from the vantage point of the vertex \((n+1\), \(n+1\), \(n+1)\). A totally symmetric plane partition is self-complementary if it is congruent (in the geometrical sense) to its complement. This cannot occur unless \(n=2m\) is even. In this paper we give several conjectures and a few theorems concerning self-complementary totally symmetric plane partitions. In particular we describe evidence which indicates a close relationship with m by m alternating sign matrices. In an earlier paper we described the close connection between m by m alternating sign matrices and descending plane partitions with no parts exceeding m. We are thus left with three classes of objects which are all apparently interrelated. There remain many unsolved problems, the simplest of which is to prove that any two of the objects have the same cardinality.
Mathematics Subject Classification ID: 05A17
zbMATH DE Number: 3995694
zbMATH Keywords: totally symmetric plane partition
zbMATH Keywords: Ferrers graph
MaRDI profile type: Publication
full work available at URL: 0097-3165(86)90098-1
OpenAlex ID: W2045968177
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cites work: Q3947818
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cites work: Alternating sign matrices and descending plane partitions
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description: scientific article; zbMATH DE number 3995694