# On the positivity of the number of t-core partitions

Acta Arithmetica (1994)

• Volume: 66, Issue: 3, page 221-228
• ISSN: 0065-1036

top

## Abstract

top
A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. 1. Let $t\ge 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n.$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,$\phantom{\rule{0.166667em}{0ex}}$4,$\phantom{\rule{0.166667em}{0ex}}$6]. If $t\ge 1$ and $n\ge 0$, then we define ${c}_{t}\left(n\right)$ to be the number of partitions of n that are t-core partitions. The arithmetic of ${c}_{t}\left(n\right)$ is studied in [3,$\phantom{\rule{0.166667em}{0ex}}$4]. The power series generating function for ${c}_{t}\left(n\right)$ is given by the infinite product: ∑n=0∞ ct(n)qn= n=1∞

## How to cite

top

Ken Ono. "On the positivity of the number of t-core partitions." Acta Arithmetica 66.3 (1994): 221-228. <http://eudml.org/doc/206601>.

@article{KenOno1994,
abstract = {A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. 1. Let $t ≥ 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n.$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,$\,$4,$\,$6]. If $t≥ 1$ and $n ≥ 0$, then we define $c_t(n)$ to be the number of partitions of n that are t-core partitions. The arithmetic of $c_t(n)$ is studied in [3,$\,$4]. The power series generating function for $c_t(n)$ is given by the infinite product: ∑n=0∞ ct(n)qn= n=1∞ },
author = {Ken Ono},
journal = {Acta Arithmetica},
keywords = {hook numbers; -core partitions; modular forms},
language = {eng},
number = {3},
pages = {221-228},
title = {On the positivity of the number of t-core partitions},
url = {http://eudml.org/doc/206601},
volume = {66},
year = {1994},
}

TY - JOUR
AU - Ken Ono
TI - On the positivity of the number of t-core partitions
JO - Acta Arithmetica
PY - 1994
VL - 66
IS - 3
SP - 221
EP - 228
AB - A partition of a positive integer n is a nonincreasing sequence of positive integers with sum $n.$ Here we define a special class of partitions. 1. Let $t ≥ 1$ be a positive integer. Any partition of n whose Ferrers graph have no hook numbers divisible by t is known as a t- core partitionof $n.$ The hooks are important in the representation theory of finite symmetric groups and the theory of cranks associated with Ramanujan’s congruences for the ordinary partition function [3,$\,$4,$\,$6]. If $t≥ 1$ and $n ≥ 0$, then we define $c_t(n)$ to be the number of partitions of n that are t-core partitions. The arithmetic of $c_t(n)$ is studied in [3,$\,$4]. The power series generating function for $c_t(n)$ is given by the infinite product: ∑n=0∞ ct(n)qn= n=1∞
LA - eng
KW - hook numbers; -core partitions; modular forms
UR - http://eudml.org/doc/206601
ER -

## References

top
1. [1] G. Andrews, EΥPHKA! num = Δ + Δ + Δ, J. Number Theory 23 (1986), 285-293.
2. [2] P. Deligne, La conjecture de Weil. I, Publ. Math. I.H.E.S. 43 (1974), 273-307.
3. [3] F. Garvan, Some congruence properties for partitions that are t-cores, Proc. London Math. Soc. 66 (1993), 449-478. Zbl0788.11044
4. [4] F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), 1-17. Zbl0721.11039
5. [5] B. Jones, The Arithmetic Theory of Quadratic Forms, Carus Math. Monographs 10, Wiley, 1950.
6. [6] A. A. Klyachko, Modular forms and representations of symmetric groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 74-85 (in Russian). Zbl0512.10019
7. [7] K. Ono, Congruences on the Fourier coefficients of modular forms on Γ₀(N), Ph.D. Thesis, The University of California, Los Angeles, 1993.
8. [8] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Publ. Math. Soc. Japan 11, Princeton Univ. Press, 1971. Zbl0221.10029

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.