Seminari del Prof. George Andrews - 7/5/2014

Per studenti

Titolo: Partitions, Compositions, and the Excitement of Ramanujan

Abstract: The theory of partitions concerns the representation of integers as distinct sums of integers. For example, the five partitions of 4 are 4, 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.

Compositions take order into account. Thus there are 8 compositions of 4, namely 4, 3 + 1, 1 + 3, 2 + 2, 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2, 1 + 1 + 1 + 1. Although seemingly more complicated, compositions are much easier to study as we shall see.

Euler was the first to study partitions seriously, and many of his discoveries are still fundamental in the subject. In this talk we introduce the basic ideas of partitions and compositions. We limit the necessary background to arithmetic and a little algebra.

The talk begins with an account of compositions. The ideas turn out to be easily understood, and the scope of the subject is easily comprehended. We then turn to partitions, the subject that the Indian genius Ramanujan revolutionized. We note several themes from Ramanujan's work suggested by our study of compositions. In each instance, we gain some appreciation of the depth and surprise of Ramanujan's insights

mercoledi' 7 maggio ore 15.30

Per docenti

Titolo: Congruences for the Fishburn Numbers
Abstract: This talk will present joint work with James Sellers. The Fishburn numbers, xi(n), have many interpretations and combinatorial applications. For example, xi(n) equals the number of upper triangular matrices with nonnegative integer entries and without zero rows or columns such that the sum of all entries equals n. In this talk I shall describe many of the interpretations of xi(n) and will prove an infinity family of congruences for xi(n). In particular, 5|xi(5n+3).

mercoledi' 7 maggio ore 18.00