# Distribution of the partition function modulo \$m\$

```@article{Ono2000DistributionOT,
title={Distribution of the partition function modulo \\$m\\$},
author={Ken Ono},
journal={Annals of Mathematics},
year={2000},
volume={151},
pages={293-307}
}```
• K. Ono
• Published 17 August 2000
• Mathematics
• Annals of Mathematics
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the…
201 Citations

### Congruence properties for the partition function

• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 2001
It is reported that such congruences are much more widespread than was previously known, and the theoretical framework that appears to explain every known Ramanujan-type congruence is described.

### ℓ-Adic properties of partition functions

• Computer Science
• 2013
This work gives a conceptual explanation of the exceptional congruence of pr observed by Boylan, as well as striking congruences of spt modulo 5, 7, and 13 recently discovered by Andrews and Garvan.

### Partition congruences and the Andrews-Garvan-Dyson crank

• K. Mahlburg
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 2005
This note announces the proof of a conjecture of Ono, which essentially asserts that the elusive crank satisfies exactly the same types of general congruences as the partition function.

### A Periodic Approach to Plane Partition Congruences

• Mathematics
Integers
• 2016
This theorem provides a novel proof of several classical congruences and establishes two new congruENCs which do not fit the scope of the theorem, using only elementary techniques, or a relationship to existing multipartition congruence.

### The spt-function of Andrews

• Mathematics
Proceedings of the National Academy of Sciences
• 2008
Using another type of identity, one based on Hecke operators, it is obtained a complete multiplicative theory for s(n) modulo 3 and congruences confirm unpublished conjectures of Garvan and Sellers.

### SUBBARAO ’ S CONJECTURE ON THE PARITY OF THE PARTITION FUNCTION

Let p(n) denote the ordinary partition function. In 1966, Subbarao [18] conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N (resp. M) ≡ r (mod t) for which

### Generalized congruence properties of the restricted partition function p(n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for

### Distribution of the partitions of n in which no part appears exactly once

In this paper we study the function g ( n ), which denotes the number of partitions of n in which no part appears exactly once. We prove that for each prime m ≥ 5, there exist Ramanujan-type

### Divisibility and distribution of 5-regular partitions

. In this paper we study the function b 3 ( n ), which denotes the number of 3-regular partitions of n , also the number of partitions where no part appears more than twice. We prove that for each

## References

SHOWING 1-10 OF 32 REFERENCES

### PERIODICITY MODULO m AND DIVISIBILITY PROPERTIES OF THE PARTITION FUNCTION(

has infinitely many solutions in non-negative integers n. This conjecture seems difficult and I have only scattered results. In ?2 of this paper it will be shown that the conjecture is true for m= 5

### Parity of the Partition Function in Arithmetic Progressions, II

• Mathematics
• 2001
Let p(n) denote the ordinary partition function. Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N = r (mod t) for which p(N) is even, and

### Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function

• M. Newman
• Mathematics
• 1957
If n is a non-negative integer, define p r(n) as the coefficient of x n in ; otherwise define p r(n) as 0. In a recent paper (2) the author established the following congruence: Let r = 4, 6, 8, 10,

### CRANKS AND T -CORES

• Mathematics
• 1990
New statistics on partitions (called cranks) are defined which combinatorially prove Ramanujan’s congruences for the partition function modulo 5, 7, 11, and 25. Explicit bijections are given for the

### Dyson's crank of a partition

• Mathematics
• 1988
holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6)

### SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13

• Mathematics
• 1967
j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular

### Defect zero p-blocks for finite simple groups

• Mathematics
• 1996
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p-block with defect 0, completing an investigation of many authors. The only finite simple groups whose

### Modular forms of weight 1/2

• Mathematics
• 1980
We will construct θ-series of weight 1/2 and 3/2 for some congruence subgroups of SL(2,ℤ) by taking appropriate coefficients of the representation \( \mathop{R}\limits^{\sim } \) of SL(2;ℝ),

### On the Parity of Additive Representation Functions

• Mathematics
• 1998
Abstract Let A be a set of positive integers,p( A , n) be the number of partitions ofnwith parts in A , andp(n)=p(N, n). It is proved that the number ofn⩽Nfor whichp(n) is even is ⪢ N , while the