On the positivity of the number of t-core partitions

Ken Ono

Acta Arithmetica (1994)

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

Abstract

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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∞

How to cite

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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

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  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

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