Hook length biases in ordinary and $t$-regular partitions Article Swipe

MAKE IMPACT

Rupam Barman , Gurinder Singh · YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2404.07485

In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular partitions. For a positive integer $k$, let $p_{(k)}(n)$ denote the number of hooks of length $k$ in all the partitions of $n$. We prove that $p_{(k)}(n)\geq p_{(k+1)}(n)$ for all $n\geq0$ and $n\ne k+1$; and $p_{(k)}(k+1)- p_{(k+1)}(k+1)=-1$ for $k\geq 2$. For integers $t\geq2$ and $k\geq1$, let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular partitions of $n$. We find generating functions of $b_{t,k}(n)$ for certain values of $t$ and $k$. Exploring hook length biases for $b_{t,k}(n)$, we observe that in certain cases biases are opposite to the biases for ordinary partitions. We prove that $b_{2,2}(n)\geq b_{2,1}(n)$ for all $n>4$, whereas $b_{2,2}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. We also propose some conjectures on biases among $b_{t,k}(n)$.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/2404.07485

PDF

https://arxiv.org/pdf/2404.07485

OA Status

green

Related Works

10

OpenAlex ID

https://openalex.org/W4394780937

Raw OpenAlex JSON

OpenAlex ID

https://openalex.org/W4394780937

Canonical identifier for this work in OpenAlex

DOI

https://doi.org/10.48550/arxiv.2404.07485

Digital Object Identifier

Title

Hook length biases in ordinary and $t$-regular partitions

Work title

Type

preprint

OpenAlex work type

Language

en

Primary language

Publication year

2024

Year of publication

Publication date

2024-04-11

Full publication date if available

Authors

Rupam Barman, Gurinder Singh

List of authors in order

Landing page

https://arxiv.org/abs/2404.07485

Publisher landing page

PDF URL

https://arxiv.org/pdf/2404.07485

Direct link to full text PDF

Open access

Yes

Whether a free full text is available

OA status

green

Open access status per OpenAlex

OA URL

https://arxiv.org/pdf/2404.07485

Direct OA link when available

Concepts

Combinatorics, Hook, Mathematics, Integer (computer science), Discrete mathematics, Computer science, Engineering, Structural engineering, Programming language

Top concepts (fields/topics) attached by OpenAlex

Cited by

0

Total citation count in OpenAlex

Related works (count)

10

Other works algorithmically related by OpenAlex

Full payload