Polynomial Sequences of Binomial-Type Arising in Graph Theory

Polynomial Sequences of Binomial-Type Arising in Graph Theory

In this paper, we show that the solution to a large class of “tiling” problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed set of polyominos on an n × n toroidal chessboard such that no two polyominos overlap is eventually a polynomial in n, and that certain sets of these polynomials satisfy binomial-type recurrences. W...

Schur Type Functions Associated with Polynomial Sequences of Binomial Type

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover this class includes examples which are useful to describe the eigenvalues of...

Polynomial Sequences of Binomial Type and Path Integrals

Polynomial sequences pn(x) of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express pn(x) as a path integral in the “phase space” N× [−π, π]. The Hamiltonian is h(φ) = ∑n=0 p ′ n(0)/n!e inφ and it produces a Schrödinger type equation for pn(x). This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an a...

A Family of Sequences of Binomial Type

For a delta operator aD − bD we find the corresponding polynomial sequence of binomial type and relations with Fuss numbers. In the case of D − 1 2D 2 we show that the corresponding Bessel–Carlitz polynomials are moments of the convolution semigroup of inverse Gaussian distributions. We also find probability distributions νt, t > 0, for which {yn(t)}, the Bessel polynomials at t, is the moment ...

On Szeged Polynomial of a Graph