Even partition functions and $2$-adic analysis

Sets with even partition functions and 2-adic integers

For P ∈ F2[z] with P (0) = 1 and deg(P ) ≥ 1, let A = A(P ) be the unique subset of N such that ∑ n≥0 p(A, n)zn ≡ P (z) (mod 2), where p(A, n) is the number of partitions of n with parts in A. Let p be an odd prime number, and let P be irreducible of order p ; i.e., p is the smallest positive integer such that P divides 1+ zp in F2[z]. N. Baccar proved that the elements of A(P ) of the form 2km...

Partition Analysis, Modular Functions, and Computer Algebra

This article describes recent developments connecting problems of enumerative combinatorics, constrained by linear systems of Diophantine inequalities, with number theory topics like partitions, partition congruences, and q-series identities. Special emphasis is put on the role of computer algebra algorithms. The presentation is intended for a broader audience; to this end, elementary introduct...

`-adic Properties of the Partition Function

Ramanujan’s famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend `-adically to 0. Although these congruences have inspired research in many directions, little is known about the `-adic behavior of these sequences for primes ` ≥ 13. We show that these sequences are governed by “fractal” behavior. Modulo any power of a prim...

p-ADIC PERIODS, p-ADIC L-FUNCTIONS, AND THE p-ADIC UNIFORMIZATION OF SHIMURA CURVES

Counting Homomorphisms and Partition Functions

Homomorphisms between relational structures are not only fundamental mathematical objects, but are also of great importance in an applied computational context. Indeed, constraint satisfaction problems, a wide class of algorithmic problems that occur in many different areas of computer science such as artificial intelligence or database theory, may be viewed as asking for homomorphisms between ...