"Theorems, problems and conjectures ",

PDF file, preprint (version 25 August, 2022).

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. [9].

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.

Attempt has been made in recording an accurate account of the results and problems in the present form of this manuscript. However, the author appreciates any relevant information on the literature, lemmas or conjectures. This page will undergo continual changes as more news come; all such matters (proper documentation, acknowledgment, progress, etc) will be made available at this site. Please indicate which version (date?) of the preprint you refer to.
Here are a few pointers for some of the conjectures and problems.
For Conjecture 1.8, we are certain that a generating function maneuvering is within reach; for an interesting combinatorial approach to emulate see ref. [4].
In Conjecture 2.1, the middle two are immediate from ref. [6] that $\sum_nx^n\prod_{u\in\lambda}\frac{t+h_u^2}{h_u^2}=\prod_j(1-x^i)^{-t-1}$; and the extreme-left and right sides are equal by duality. The main wish here is to prove the rest; a combinatorial argument is strongly desired.
For Problem 5.3, see ref. [7] for a possible technique on a similar result.
Conjecture 8.1 is amenable to generating function manipulation; yet, a combinatorial interpretation is more interesting (ref. [5] is recommended).
Related papers
A multiset hook length formula and some applications.

Generalizations of Nekrasov-Okounkov Identity.

Euler-Mahonian statistics via polyhedral geometry.

Comments and updates
Amer Iqbal believes a q-deformed version of Cor. 5.2 can also be obtained using ideas from topological strings (no specifics provided) - July 18, 2012.

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.

Conjectures 11.1-11.5, 11.9, 12.2 have found resolutions and extensions:
Tewodros Amdeberhan, Emily Leven, Multi-cores, posets, and lattice paths.

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.

Bernhard Heim, Markus Neuhauser On conjectures regarding the Nekrasov-Okounkov hook length formula, proved Conjecture 2.1.

Noah Kravitz On the number of simultaneous core partitions with d-distinct parts.

Jineon Baek, Hayan Nam, Myungjun Yu A bijective proof of Amdeberhan's conjecture on the number of (s,s+2)-core partitions with distinct parts.

Armin Straub Core partitions into distinct parts and an analog of Euler's theorem.


Earlier versions of the manuscript benefited from useful suggestions by M Can, M Joyce, R P Stanley.



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