# Refined Enumeration of Corners in Tree-like Tableaux

## TL;DR

This paper confirms two conjectures on the refined counting of non-occupied corners in tree-like tableaux and their symmetric variants, using intermediate combinatorial structures linked to permutation and alternative tableaux.

## Contribution

It provides a proof for conjectures related to corner enumeration in tree-like tableaux and introduces methods connecting various combinatorial structures.

## Key findings

- Confirmed conjectures on corner enumeration in tree-like tableaux
- Established links between different combinatorial structures
- Enhanced understanding of symmetric and non-symmetric tableau enumeration

## Abstract

Tree-like tableaux are certain fillings of Ferrers diagrams originally introduced by Aval et al., which are in simple bijections with permutation tableaux coming from Postnikov's study of totally nonnegative Grassmanian and alternative tableaux introduced by Viennot. In this paper, we confirm two conjectures of Gao et al. on the refined enumeration of non-occupied corners in tree-like tableaux and symmetric tree-like tableaux via intermediate structures of alternative tableaux, linked partitions, type $B$ alternative tableaux and type $B$ linked partitions.