03/02/2020, 06:28 AM
I recommend checking out Analytic Combinatorics and John Baez's writings.
Umbral calculus recognizes the reality that we live in a polynomial-centric world in math, a concept aligned with n-dimensional hypercubes. But there are relationships that span the hypercube and other basic combinatorial structures. Rota reduced this to linear operator, so the weird looking efficacy of umbral calculus is legit.
Our common interest is iterated functions. Well in combinatorial terms we want to look into the realm of compositions instead of polynomials. One important property is that recursive compositions terminate at https://oeis.org/A000311 instead of continuing to grow. The combinatorial structure A000311 is "Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n". It is also the number of ways to classify a group of objects.
Umbral calculus recognizes the reality that we live in a polynomial-centric world in math, a concept aligned with n-dimensional hypercubes. But there are relationships that span the hypercube and other basic combinatorial structures. Rota reduced this to linear operator, so the weird looking efficacy of umbral calculus is legit.
Our common interest is iterated functions. Well in combinatorial terms we want to look into the realm of compositions instead of polynomials. One important property is that recursive compositions terminate at https://oeis.org/A000311 instead of continuing to grow. The combinatorial structure A000311 is "Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n". It is also the number of ways to classify a group of objects.
Daniel