On the positivity of the number of t-core partitions

• Autorzy: Ken Ono
• Języki publikacji: EN

Abstrakty

EN

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. \de{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 partition} of $n.$ \vskip 4pt plus 2pt 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}^{∞} c_t(n)q^n= \prod_{n=1}^{∞

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1994
• Tom: 66
• Numer: 3
• Strony: 221-228

Daty

wydano

1994

otrzymano

1993-04-06

poprawiono

1993-10-12

Twórcy

autor

Ken Ono

• Department of Mathematics, the University of Georgia, Athens, Georgia 30602, U.S.A.

Bibliografia

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• [3] F. Garvan, Some congruence properties for partitions that are t-cores, Proc. London Math. Soc. 66 (1993), 449-478.
• [4] F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), 1-17.
• [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).
• [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.