The Ramanujan Journal

, Volume 18, Issue 1, pp 103–112

# Arithmetical properties of the number of t-core partitions

Article

## Abstract

Let Λ={λ 1⋅⋅⋅λ s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ i nodes in the ith row. Let λ j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ i +λ j ′−ij+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.

## Keywords

t-core partition Congruence Modular form

11P83

## References

1. 1.
Andrews, G.: The Theory of Partitions. Cambridge University Press, New York (1984)
2. 2.
Andrews, G., Bessenrodt, C., Olsson, J.: Partition identities and labels for some modular characters. Trans. Am. Math. Soc. 344, 597–615 (1994)
3. 3.
Boylan, M.: Congruences for 2t-core partition functions. J. Number Theory 92(1), 131–138 (2002)
4. 4.
Brauer, R.: Representations of finite simple groups. In: Lecture Notes on Modern Math., vol. 1, pp. 133–175. Wiley, New York (1963) Google Scholar
5. 5.
de Robinson, G.: Representation Theory of the Symmetric Group. Toronto Univ. Press, Toronto (1961)
6. 6.
Fong, P., Srinivasan, B.: The blocks of finite general linear groups and unitary groups. Invent. Math. 69, 109–153 (1982)
7. 7.
Garvan, F.: Some congruence properties for partition that are p-cores. Proc. Lond. Math. Soc. 66, 449–478 (1993)
8. 8.
Garvan, F., Kim, D., Stanton, D.: Cranks and t-core. Invent. Math. 101, 1–17 (1990)
9. 9.
Gordon, B., Hughes, K.: Multiplicative properties of η products II. Contemp. Math. 143, 415–430 (1993)
10. 10.
Granville, A., Ono, K.: Defect zero p-blocks for finite simple groups. Trans. Am. Math. Soc. 348, 331–347 (1996)
11. 11.
Hirschhorn, M., Sellers, J.A.: Some amazing facts about 4-cores. J. Number Theory 60(1), 51–69 (1996)
12. 12.
Hirschhorn, M., Sellers, J.A.: Some parity results for 16-cores. Ramanujan J. 3, 281–296 (1999)
13. 13.
Isaacs, M.: Character Theory of Finite Simple Groups. Academic, New York (1976) Google Scholar
14. 14.
15. 15.
Kolitsch, L.W., Sellers, J.A.: Elementary proofs of infinitely many congruences for 8-cores. Ramanujan J. 3(2), 221–226 (1999)
16. 16.
Ligozat, G.: Courbes modulaires de genre 1. Bull. Soc. Math. France 43, 1–80 (1975)
17. 17.
Mahlburg, K.: More congruences for the coefficients of quotients of Eisentein series. J. Number Theory 115, 89–99 (2005)
18. 18.
Niwa, S.: Modular forms of half integral weight and integral weight of certain theta functions. Nagoya Math. J. 56, 147–161 (1974)
19. 19.
Newman, M.: Construction and application of a certain class of modular functions II. Proc. Lond. Math. Soc. 9, 353–387 (1959)
20. 20.
Ono, K.: The distribution of the partition function modulo m. Ann. Math. 151, 293–307 (2000)
21. 21.
Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series. CBMS Regional Conference Series in Mathematics, vol. 102. Am. Math. Soc., Providence (2004)
22. 22.
Ono, K., Sze, L.: 4-core partitions and class number. Acta Arith. 80(3), 249–272 (1997)
23. 23.
Ramanujan, S.: Congruence properties of partitions. Proc. Lond. Math. Soc. 19(2), 207–210 (1919)
24. 24.
Serre, J.-P.: Divisibilité de certaines fonctions arithmétiques. Enseign. Math. 22, 227–260 (1976)
25. 25.
Shimura, G.: On modular forms of half-integral weight. Ann. Math. 97, 440–481 (1973)
26. 26.
Tate, J.: The non-existence of certain Galois extensions of Q unramified outside 2. In: Arithmetic Geometry, Tempe, AZ, 1993. Contemporary Mathematics, vol. 174, pp. 153–156. Am. Math. Soc., Providence (1994) Google Scholar