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A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

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Fully Complementary Higher Dimensional Partitions.

Florian Schreier-Aigner¹

¹University of Vienna, Wien, Austria.

Annals of Combinatorics

|March 25, 2025

Summary

This summary is machine-generated.

We introduce fully complementary higher dimensional partitions (FCPs) and their generating function. This study also explores new symmetry classes for plane partitions, including quasi-transpose-complementary plane partitions.

Keywords:

Higher dimensional partitions MacMahon Plane partitions Symmetry classes

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Area of Science:

Combinatorics
Algebraic Combinatorics
Partition Theory

Background:

Symmetry classes of plane partitions are fundamental in combinatorial research.
Higher dimensional partitions generalize plane partitions but are less studied.
Existing research lacks a systematic study of higher dimensional partition symmetries.

Purpose of the Study:

Introduce a new symmetry class: fully complementary higher dimensional partitions (FCPs).
Derive a generating function formula for FCPs.
Explore variations of classical symmetry classes for plane partitions using FCPs.

Main Methods:

Develop theoretical framework for higher dimensional partitions.
Utilize generating function techniques.
Analyze symmetry properties in dimension 2.

Main Results:

Established the definition and properties of fully complementary higher dimensional partitions (FCPs).
Derived a formula for the generating function of FCPs.
Conjectured three new symmetry classes for plane partitions.
Proved that quasi-transpose-complementary plane partitions are equinumerous to symmetric plane partitions.

Conclusions:

The introduction of FCPs expands the study of higher dimensional partitions.
The findings provide new insights into the structure of plane partitions.
This work opens avenues for further research in combinatorial symmetries.