Let p(n) denote the number of unrestricted partitions of n. For i = 0, 2, let pi(n) denote the number of partitions π of n such that O(π)−O(π) ≡ i (mod 4). Here O(π) denotes the number of odd parts of the partition π and π is the conjugate of π. R. Stanley [13], [14] derived an infinite product representation for the generating function of p0(n)− p2(n). Recently, H. Swisher [15] employed the ci...

Ramanujan also conjectured that congruences (1) exist for the cases A = 5 , 7 , or 11 . This conjecture was proved by Watson [17] for the cases of powers of 5 and 7 and Atkin [3] for the cases of powers of 11. Since then, the problem of finding more examples of such congruences has attracted a great deal of attention. However, Ramanujan-type congruences appear to be very sparse. Prior to the la...

Markov Random Fields (MRFs) have proven very powerful both as density estimators and feature extractors for classification. However, their use is often limited by an inability to estimate the partition function Z. In this paper, we exploit the gradient descent training procedure of restricted Boltzmann machines (a type of MRF) to track the log partition function during learning. Our method reli...

In this paper we discuss efficient algorithms for computing the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjectures related to the partition function. We present the distribution of p(N) for N ≤ 109 for primes up to 103 and small powers of 2 and 3.

When belief propagation (BP) converges, it does so to a stationary point of the Bethe free energy F , and is often strikingly accurate. However, it may converge only to a local optimum or may not converge at all. An algorithm was recently introduced by Weller and Jebara for attractive binary pairwise MRFs which is guaranteed to return an !-approximation to the global minimum of F in polynomial ...