# Number Theory Seminar

Presented by: Karl Dilcher

Title: “On the polynomial part of a restricted partition function”

Abstract: Given a finite set d of distinct positive

integers, the restricted partitionfunction W(d,n) counts the number of ways of writing a positive integer n as a sum of elements from d, disregarding order of the summands. Sylvester showed that for a fixed d, W(d,n) can be written as a finite sum of functions he called "waves", the most basic and important one being a polynomial. I will present an explicit formula for this polynomial part, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to Raabe's well-known multiplication formula for the ordinary Bernoulli polynomials. As a consequence of the main result we obtain an asymptotic expression of the first Sylvester wave as the coefficients of the restricted partition grow arbitrarily large. As part of this talk I will introduce and discuss a symbolic notation for Bernoulli numbers and polynomials.(Joint work with Christophe Vignat).

#### Category

Lectures, Seminars

#### Time

Starts:

Ends:

#### Location

Seminar Room #227 Chase Building

#### Cost

Free

#### Contact

Ellen Lynch