Welcome to the Hook Formula Homepage!

Hook Length Formulas for Partitions and Plane Trees

Guo-Niu Han (Strasbourg)

This web site is devoted to my recent works on hook length formulas for partitions and plane trees.

My papers on hook length formulas

[1] Discovering hook length formulas by an expansion technique, Elec. J. Combin. Vol. 15(1), R133, 41 pages, 2008. [Abstract and Download]

[2] New hook length formulas for binary trees, Combinatorica, in press, 4 pages, 2008. [Abstract and Download] arXiv:0804.3638 [math.CO]

[3] Yet another generalization of Postnikov's hook length formula for binary trees, SIAM J. Discrete Math, in press, 4 pages, 2008. [Abstract and Download] arXiv:0804.4268 [math.CO]

[4] Some conjectures and open problems on partition hook lengths, Experimental Mathematics, in press, 15 pages, 2008. [Abstract and Download]

[5] The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, 28 pages, 2008. [Abstract and Download] arXiv:0805.1398 [math.CO]

[5'] An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, 35 pages, 2008. [Abstract and Download] (unpublished, arXiv:0804.1849 [math.CO])

[5''] A generalized hook formula for partitions via t-cores and its applications, 13 pages, 2008. (unpublished, draft version)

[6] (with Ken Ono) Hook lengths and 3-cores, Annals of Combin., in press, 7 pages, 2008. [Abstract and Download] arXiv:0805.2461 [math.NT]

[7] Hook lengths and shifted parts of partitions, 9 pages, 2008. [Abstract and Download] arXiv:0807.1801 [math.CO]

Maple programs for verifying the formulas and conjectures

HookExp.mpl - Document [1]

Last news and comments

[2008.05.01] Laura Yang has generalized hook length formulas for binary trees in [2] and [3] to k-ary trees, arXiv:0805.0109 [math.CO]

[2008.05.04] Mihai Cipu has proved Conjecture 5.2 in [4]. Private communication.

[2008.05.05] Richard Stanley has found an elementary, but not bijective, proof of the marked hook formula in [4] and [5']. Private communication.

[2008.05.07] Bruce Sagan has found probabilistic proofs of hook length formulas for binary trees in [2], arXiv:0805.0817 [math.CO]

[2008.05.08] Why papers [5'] and [5''] remain unpublished ?

[2008.05.10] Richard Stanley can prove the k=2 case of Conjecture 3.1 in [4]. Private communication.

[2008.06.18] Richard Stanley proves Conjecture 3.1 in [4]. Private communication.

[2008.07.01] Answer the Quiz.

[2008.07.02] Richard Stanley proved and generalized Conjecture 3.1 in [4] arXiv:0807.0383 [math.CO]

[2008.07.11] Gil Kalai wrote some comments about my recent works on hook length formula in his blog, Powers of Euler Products and Han's Marked Hook Formula

[2008.07.17] Paper [4] is cited by Tewodros Amdeberhan, Differential operators, shifted parts, and hook lengths, arXiv:0807.2473 [math.CO]

[2008.07.21] Emily Clader, Yvonne Kemper, Matt Wage, Lacunarity of certain partition theoretic generating functions arising from Han's generalization of the Nekrasov-Okounkov formula

[2008.07.21] Ameya Velingker, An exact formula for the coefficients of Han's generating function

[2008.08.04] Kevin Carde, Joe Loubert, Aaron Potechin, Adrian Sanborn, Proof of Han's Hook Expansion Conjecture

Quiz

Triple mixed hook formula: What is the numerical value of the following expression ? 0, 1 ,2, 3 or 1001 ?

where t ranges over all binary trees with 1001 vertices, h_v is the hook length for trees and h_u is the hook length for partitions.

Answer : click here ...

Back to List of Papers - Send comments to Email - First version: 2008/04/04 - Last update: 2008/09/10