Gončarov polynomials in partition lattices and exponential families. (English) Zbl 1462.05013
Summary: Classical Gončarov polynomials arose in numerical analysis as a basis for the solutions of the Gončarov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, R. Lorentz et al. [in: Connections in discrete mathematics. A celebration of the work of Ron Graham. Cambridge: Cambridge University Press. 56–85 (2018; Zbl 1400.41002)] introduced the sequence of generalized Gončarov polynomials associated to a pair \((\Delta, \mathcal{Z})\) of a delta operator \(\Delta\) and an interpolation grid \(\mathcal{Z} \). Generalized Gončarov polynomials share many nice algebraic properties and have a connection with the theories of binomial enumeration and order statistics. In this paper we give a complete combinatorial interpretation for any sequence of generalized Gončarov polynomials. First we show that they can be realized as weight enumerators in partition lattices. Then we give a more concrete realization in exponential families and show that these polynomials enumerate various enriched structures of vector parking functions.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05A18 | Partitions of sets |
| 06A07 | Combinatorics of partially ordered sets |
Citations:
Zbl 1400.41002Software:
OEISReferences:
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