# Hooks and powers of parts in partitions

@article{Bacher2001HooksAP, title={Hooks and powers of parts in partitions}, author={Roland Bacher and Laurent Manivel}, journal={S{\'e}minaire Lotharingien de Combinatoire}, year={2001}, volume={47}, pages={11} }

This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th parts in partitions of n.

## 16 Citations

An involution for symmetry of hook length and part length of pointed partitions

- Mathematics, Computer ScienceDiscret. Math.
- 2010

An involution for symmetry of hook length and part length of partitions

- Mathematics
- 2009

A {\em pointed partition} of $n$ is a pair $(\lambda, v)$ where $\lambda\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook…

Symmetry distribution between hook length and part length for partitions

- Computer Science, MathematicsDiscret. Math.
- 2009

On t-Core Towers and t-Defects of Partitions

- Mathematics
- 2015

We study generating functions which count the sizes of t-cores of partitions, and, more generally, the sizes of higher rows in t-core towers. We then use these results to derive an asymptotic results…

Colored partitions and the hooklength formula: partition statistic identities

- Mathematics
- 2018

We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by…

Combining hook length formulas and BG-ranks for partitions via the Littlewood decomposition

- Mathematics
- 2011

Recently, the first author has studied hook length formulas for partitions in a systematic manner. In the present paper we show that most of those hook length formulas can be generalized and include…

HOOK TYPE TABLEAUX AND PARTITION IDENTITIES

- 2020

In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of the Stanley’s and Elder’s theorem without the use of partition statistics in…

Discovering Hook Length Formulas by an Expansion Technique

- Computer Science, MathematicsElectron. J. Comb.
- 2008

A hook length expansion technique is introduced and how to discover old and new hook length formulas for partitions and plane trees are explained, expained by well-known combinatorial arguments.

The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

- Mathematics
- 2008

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the…

Quasimodularity and large genus limits of Siegel-Veech constants

- Mathematics
- 2016

Quasimodular forms were first studied in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms.…

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