Abstract
Univariate Gončarov polynomials arose from the Gončarov interpolation problem in numerical analysis. They provide a natural basis of polynomials for working with u-parking functions, which are integer sequences whose order statistics are bounded by a given sequence u. In this paper, we study multivariate Gončarov polynomials, which form a basis of solutions for multivariate Gončarov interpolation problem. We present algebraic and analytic properties of multivariate Gončarov polynomials and establish a combinatorial relation with integer sequences. Explicitly, we prove that multivariate Gončarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in ℕk. It leads to a higher-dimensional generalization of parking functions, for which many enumerative results can be derived from the theory of multivariate Gončarov polynomials.
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Khare, N., Lorentz, R. & Yan, C.H. Bivariate Gončarov polynomials and integer sequences. Sci. China Math. 57, 1561–1578 (2014). https://doi.org/10.1007/s11425-014-4827-x
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DOI: https://doi.org/10.1007/s11425-014-4827-x