Enumeration of Cylindric Plane Partitions - part I

TL;DR

This paper introduces a $(q,t)$-analog of Borodin's enumeration identity for cylindric plane partitions, extending previous work and providing a combinatorial interpretation via lattice paths.

Contribution

It extends Borodin's generating series to a $(q,t)$-analog using vertex operators and Macdonald polynomials, and offers a new combinatorial interpretation.

Findings

Derived a $(q,t)$-analog of Borodin's identity

Connected Macdonald weights to lattice path models

Extended enumeration techniques for cylindric plane partitions

Abstract

Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. The first result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. Our proof uses commutation relations for $(q,t)$-vertex operators acting on Macdonald polynomials as given by Garsia, Haiman and Tesla. The second result of this paper is an explicit combinatorial interpreation of the $(q,t)$-Macdonald weight in terms of a non-intersecting lattice path model on the cylinder.