Expected Frobenius numbers

Euler–frobenius Numbers and Rounding

We study the Euler–Frobenius numbers, a generalization of the Eulerian numbers, and the probability distribution obtained by normalizing them. This distribution can be obtained by rounding a sum of independent uniform random variables; this is more or less implicit in various results and we try to explain this and various connections to other areas of mathematics, such as spline theory. The mea...

Bounds on generalized Frobenius numbers

Let N ≥ 2 and let 1 < a1 < · · · < aN be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that has no representation as PN i=1 aixi where x1, ..., xN are nonnegative integers. More generally, the s-Frobenius number is defined to be the largest positive integer that has precisely s distinct representations like this. We use techniques...

Faster Algorithms for Frobenius Numbers

The Frobenius problem, also known as the postage-stamp problem or the moneychanging problem, is an integer programming problem that seeks nonnegative integer solutions to x1a1 + · · · + xnan = M , where ai and M are positive integers. In particular, the Frobenius number f(A), where A = {ai}, is the largest M so that this equation fails to have a solution. A simple way to compute this number is ...

Expected Crossing Numbers

The expected value for the weighted crossing number of a randomly weighted graph is studied. A variation of the Crossing Lemma for expectations is proved. We focus on the case where the edge-weights are independent random variables that are uniformly distributed on [0, 1].

Limiting behaviour of large Frobenius numbers