The theory of binomial enumeration is variously called the calculus of finite differences or the umbral calculus. This theory studies the analogies between various sequences of polynomials and the powers sequence . The subscript n in was thought of as the shadow ( ``umbra'' means ``shadow'' in Latin, whence the name umbral calculus) of the superscript n in , and many parallels were discovered between such sequences.
Take the example of the lower factorial polynomials . Just as counts the number of functions from an n-element set to an x-element set, counts the number of injections. Just as the derivative maps to , the forward difference operator maps to . Just as also polynomials can be expressed in terms of via Taylor's theorem
Newton's theorem allows similar expressions for
where . Just as is expanded using the binomial theorem
expands by Vandermonde's identity
And so on. [67,60]
This theory is quite classical with its roots in the works of Barrow and Newton---expressed in the belief the some polynomial sequences such as really were just like the powers of x. Nevertheless, many doubts arose as to the correctness of such informal reasoning, despite various (see e.g., ) attempts to set it on an axiomatic base.
The contribution of Rota's school was to first set umbral calculus on a firm logical foundation by using operator methods [67,106]. That being done, sequences of polynomials of binomial type
could be for once studied systematically rather than as a collection of isolated yet philosophically similar results. The sister sequence of divided powers then obeys the identity
Given any species of combinatorial structures (or quasi-species), let be the number of functions from an n-element set to an x-element set enriched by this species. A function is enriched by associating a (weighted) structure with each of its fibers. All sequences of binomial type arise in this manner, and conversely, all such sequence are of binomial type.