(This post is mostly intended for my own reference, as I found myself repeatedly looking up several conversions between polynomial bases on various occasions.)

Let denote the vector space of polynomials of one variable with real coefficients of degree at most . This is a vector space of dimension , and the sequence of these spaces form a filtration:

A standard basis for these vector spaces are given by the monomials : every polynomial in can be expressed uniquely as a linear combination of the first monomials . More generally, if one has any sequence of polynomials, with each of degree exactly , then an easy induction shows that forms a basis for .

In particular, if we have two such sequences and of polynomials, with each of degree and each of degree , then must be expressible uniquely as a linear combination of the polynomials , thus we have an identity of the form

for some change of basis coefficients . These coefficients describe how to convert a polynomial expressed in the basis into a polynomial expressed in the basis.

Many standard combinatorial quantities involving two natural numbers can be interpreted as such change of basis coefficients. The most familiar example are the binomial coefficients , which measures the conversion from the shifted monomial basis to the monomial basis , thanks to (a special case of) the binomial formula:

thus for instance

More generally, for any shift , the conversion from to is measured by the coefficients , thanks to the general case of the binomial formula.

But there are other bases of interest too. For instance if one uses the falling factorial basis

then the conversion from falling factorials to monomials is given by the Stirling numbers of the first kind :

thus for instance

and the conversion back is given by the Stirling numbers of the second kind :

thus for instance

If one uses the binomial functions as a basis instead of the falling factorials, one of course can rewrite these conversions as

and

thus for instance

and

As a slight variant, if one instead uses rising factorials

then the conversion to monomials yields the unsigned Stirling numbers of the first kind:

thus for instance

One final basis comes from the polylogarithm functions

For instance one has

and more generally one has

for all natural numbers and some polynomial of degree (the Eulerian polynomials), which when converted to the monomial basis yields the (shifted) Eulerian numbers

For instance

These particular coefficients also have useful combinatorial interpretations. For instance:

The binomial coefficient is of course the number of -element subsets of .

is of course the number of -element subsets of . The unsigned Stirling numbers of the first kind are the number of permutations of with exactly cycles. The signed Stirling numbers are then given by the formula .

of the first kind are the number of permutations of with exactly cycles. The signed Stirling numbers are then given by the formula . The Stirling numbers of the second kind are the number of ways to partition into non-empty subsets.

of the second kind are the number of ways to partition into non-empty subsets. The Eulerian numbers are the number of permutations of with exactly ascents.

These coefficients behave similarly to each other in several ways. For instance, the binomial coefficients obey the well known Pascal identity

(with the convention that vanishes outside of the range ). In a similar spirit, the unsigned Stirling numbers of the first kind obey the identity

and the signed counterparts obey the identity

The Stirling numbers of the second kind obey the identity

and the Eulerian numbers obey the identity