Skew doubled shifted plane partitions: calculus and asymptotics

TL;DR

This paper introduces a new summation formula for Schur processes and applies it to derive generating functions and asymptotics for doubled shifted plane partitions and symmetric cylindric partitions, advancing combinatorial enumeration methods.

Contribution

It establishes a novel complete summation formula for Schur processes and applies it to obtain new enumerative and asymptotic results for specific classes of plane partitions.

Findings

Derived the generating function for doubled shifted plane partitions.

Established the asymptotic formula depending on diagonal width.

Extended methods to symmetric cylindric partitions.

Abstract

Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuleti\'c, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.