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Enumerating coloured partitions in 2 and 3 dimensions

Part of: Combinatorics

Published online by Cambridge University Press:  19 July 2019

BEN DAVISON ,
JARED ONGARO  and
BALÁZS SZENDRŐI
BEN DAVISON
Affiliation:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, KING’s Buildings, Edinburgh, EH9 3FD. e-mail: ben.davison@ed.ac.uk
JARED ONGARO
Affiliation:
School of Mathematics, University of Nairobi, Chiromo Campus, P.O. Box 30197, 00100 Nairobi, Kenya. e-mail: ongaro@uonbi.ac.ke
BALÁZS SZENDRŐI
Affiliation:
Mathematical Institute, University of Oxford, Andrews Wiles Building, Woodstock Road, Oxford, OX2 699. e-mail: szendroi@maths.ox.ac.uk
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Abstract

We study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.

MSC classification

Primary: 05A17: Partitions of integers
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 3 , November 2020 , pp. 479 - 505
Copyright
© Cambridge Philosophical Society 2019

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