Approximate counting of standard set-valued tableaux

TL;DR

This paper introduces randomized algorithms for efficiently sampling and approximately counting standard set-valued tableaux, extending existing methods and providing new tools for algebraic combinatorics.

Contribution

It develops the first fully polynomial almost uniform sampler and approximation scheme for certain classes of standard set-valued tableaux shapes.

Findings

Developed a FPAUS for asymptotically rank two partitions.

Constructed a FPRAS for counting tableaux of specific shapes.

Extended existing algorithms to broader classes of combinatorial objects.

Abstract

We present a randomized algorithm for generating standard set-valued tableaux by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of asymptotically rank two partitions, we use this algorithm to give a fully polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux. This FPAUS is then used to construct a fully polynomial randomized approximation scheme (FPRAS) for counting the number of standard set-valued tableaux for such shapes. We also construct a FPAUS and FPRAS for standard set-valued tableaux when either the size of the partition or the difference between the maximum value and the size of the partition is fixed. Our methods build on the work of Jerrum-Valiant-Vazirani and provide a framework for constructing FPAUS's and FPRAS's for other counting problems in algebraic combinatorics.