We assume basic knowledge on Eulerian polynomials (including q-analogues),
integer partitions, Young diagrams, symmetric functions, group actions,
and a bit of representation theory (e.g. the role of Schur functions);
see ref. .
These notes are designed to offer some codicils (perhaps new) to related work, a list of problems and conjectures seeking (preferably) combinatorial proofs. The main items are Eulerian polynomials, Burnside lemma, and hook/contents of a Young diagram, mostly on the latter. New additions include items on Frobenius theorem and t-core partitions; most recently, some problems on (what we call) colored overpartitions. Formulas analogues to or in the spirit of works by Han, Nekrasov-Okounkov, Stanley and Polya are distributed throughout.
Generalizations of Nekrasov-Okounkov Identity.
Euler-Mahonian statistics via polyhedral geometry.
Anna R B Fan, Harold R L Yang, Rebecca T Yu On the Maximum Number of k-Hooks of Partitions of n have generalized and proved Problem 2.2 - Dec. 14, 2012.
Jane Y X Yang, Michael X X Zhong, Robin D P Zhou On the Enumeration of $(s,s+1,s+2)$-Core Partitions.
Huan Xiong The number of simultaneous core partitions.
Huan Xiong On the largest size of $(t,t+1,..., t+p)$-core partitions.
Rishi Nath Symmetry in maximal $(s-1,s+1)$ cores.
Rishi Nath & James A. Sellers A combinatorial proof ... between (2k-1,2k+1)-cores & (2k-1,2k,2k+1)-cores.
Amol Aggrawal When does the set of (a,b,c)-core partitions have unique max element?
Conjecture 12.2 is now a theorem (joint G. Andrews, in preparation).
Victor Wang Simultaneous core partitions: parameterizations and sums, proved Conjecture 11.5.
Paul Johnson proved Conjecture 11.5 (private correspondence).
T Amdeberhan, M Apagodu, D Zeilberger Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, solved Problem 11.6.
Huan Xiong Core partitions with distinct parts, proves Conjecture 11.9; Armin Straub (private correspondence) proves Conjecture 11.9(a), independently.
Robin DaPao Zhou The Raney Numbers and (s,s+1)-Core Partitions, proved Conjecture 11.7.
Earlier versions of the manuscript benefited from useful suggestions by M Can, M Joyce, R P Stanley.