Theorems, Problems and Conjectures

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

15 pages

References in corpus (4)

• An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths
• Proof of Han's Hook Expansion Conjecture
• Results and conjectures on simultaneous core partitions
• On the Maximum Number of k-Hooks of Partitions of n

Cited by in corpus (17)

• Explicit expressions for the moments of the size of an (s,s+1)-core partition with distinct parts
• Core partitions with distinct parts
• The number of simultaneous core partitions
• On the largest size of $(t,t+1,...,t+p)$-core partitions
• A bijective proof of Amdeberhan's conjecture on the number of $(s,s+2)$-core partitions with distinct parts
• Sizes of Simultaneous Core Partitions
• Explicit (Polynomial!) Expressions for the Expectation, Variance and Higher Moments of the Size of a (2n + 1, 2n + 3)-core partition with Distinct Parts
• On the Enumeration of $(s,s+1,s+2)$-Core Partitions
• Bijections between t-Core Partitions and t-Tuples
• Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle
• Modular Nekrasov-Okounkov formulas
• On the number of simultaneous core partitions with $d$-distinct parts
• The Raney Numbers and $(s,s+1)$-Core Partitions
• On the Maximum Number of k-Hooks of Partitions of n
• Cores with distinct parts and bigraded Fibonacci numbers
• On the polynomiality and asymptotics of moments of sizes for random $(n,dn\pm 1)$-core partitions with distinct parts
• Core Partitions With d-Distinct Parts