Theorems, Problems and Conjectures

TL;DR

This paper discusses various combinatorial problems, conjectures, and formulas related to Eulerian polynomials, Young diagrams, multi-core partitions, and chromatic polynomials, aiming to inspire further research and proofs.

Contribution

It introduces new problems and conjectures in combinatorics, especially on chromatic polynomials and related structures, expanding on existing theories and formulas.

Findings

Proposes new combinatorial problems and conjectures.

Provides formulas related to Eulerian polynomials and Young diagrams.

Highlights connections to existing theorems and recent research.

Abstract

These notes are designed to offer some (perhaps new) codicils to related work, a list of problems and conjectures seeking (preferably) combinatorial proofs. The main items are Eulerian polynomials and hook/contents of Young diagram, mostly on the latter. We also have items on Frobenius theorem and multi-core partitions; most recently, some problems on (what we call) colored over-partitions. Formulas analogues to or in the spirit of works by Han, Nekrasov-Okounkov and Stanley are distributed throughout. Concluding remarks are provided at the end in hopes of directing the interested researcher, properly. The newly added problem is on chromatic polynomials