MacMahon's partition analysis XIII: Schmidt type partitions and modular forms

Integer partitions, probabilities and quantum modular forms

What is the probability that the smallest part of a random integer partition of N is odd? What is the expected value of the smallest part of a random integer partition of N? It is straightforward to see that the answers to these questions are both 1, to leading order. This paper shows that the precise asymptotic expansion of each answer is dictated by special values of an arithmetic L-function....

Mock modular forms and d-distinct partitions

Article history: Received 22 August 2013 Accepted 20 December 2013 Available online 17 January 2014 Communicated by George E. Andrews MSC: 11P82 11P84 11F37 11F50 33D15

The Partition Function and Modular Forms

1. Intro to partition function and modular forms 1 2. Partition function leading term, without modular forms 2 3. Modular form basics 5 4. First application: Rademacher’s formula 6 4.1. A Transformation Formula for the η Function 6 4.2. Rademacher’s Convergent Series 14 5. Second application: Ramanujan congruences 22 5.1. Additional Results from Modular Functions 22 5.2. Proof of Ramanujan Cong...

Macmahon's Partition Analysis Xi: the Search for Modular Forms

Partition Analysis Xii: Plane Partitions