Theorems, Problems and Conjectures

2012/07/17 by Tewodros Amdeberhan, Amdeberhan, Tewodros · 6 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research

paper · pdf · doi:10.48550/arxiv.1207.4045

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

Cited by

• A bijective proof of Amdeberhan's conjecture on the number of (s, s+2)-core partitions with distinct parts
• Core partitions with distinct parts
• The Raney Numbers and (s,s+1)-Core Partitions
• Bijections between t-Core Partitions and t-Tuples
• On the largest size of (t,t+1,..., t+p)-core partitions
• On the Maximum Number of k-Hooks of Partitions of n

Related

• From Poset Topology to q-Eulerian Polynomials to Stanley's Chromatic Symmetric Functions
• Chromatic Polynomial and Heaps of Pieces
• Enumeration of diagonally colored Young diagrams
• A Contribution to the Theory of Chromatic Polynomials
• q-Chromatic polynomials