This talk will explore an unexpected connection between a family of graphs, called threshold graphs, and two seemingly  unrelated families of integer sequences — one arising from enumerative combinatorics and the other from commutative algebra. I will introduce these three families of objects, present an explicit  one-to-one-to-one correspondence between them, and show how a  well-studied random model for threshold graphs can be used to answer  probabilistic questions in combinatorics and commutative algebra regarding expected values and properties of the integer sequences.  No prior knowledge of commutative algebra will be assumed; all of  the essential ideas will be illustrated with concrete examples.  This is joint work with Alexander Engstrom and Christian Go.