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

Acta Arithmetica (1994)

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

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

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