# Dual graphs from noncommutative and quasisymmetric Schur functions

## TL;DR

This paper explores the combinatorial structures of compositions related to noncommutative and quasisymmetric Schur functions, revealing dual graph structures and their deformations, advancing understanding in algebraic combinatorics.

## Contribution

It establishes dual graded and filtered graph structures for composition posets derived from Pieri rules of noncommutative and quasisymmetric Schur functions, connecting these frameworks.

## Key findings

- Posets from Pieri rules can be endowed with dual graded graph structures.
- Posets can also be structured as dual filtered graphs.
- The work links noncommutative and quasisymmetric Schur functions through these graph structures.

## Abstract

By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered graphs when paired with the poset of compositions arising from the Pieri rules for quasisymmetric Schur functions and its deformation.