Science China Mathematics

, Volume 57, Issue 8, pp 1561–1578

# Bivariate Gončarov polynomials and integer sequences

• Niraj Khare
• Rudolph Lorentz
• Catherine Huafei Yan
Articles

## 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 ℕ k . 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.

### Keywords

Gončarov polynomials interpolation parking functions order statistics

### MSC(2010)

05A15 05A10 41A05 41A63

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© Science China Press and Springer-Verlag Berlin Heidelberg 2014

## Authors and Affiliations

• Niraj Khare
• 1
• Rudolph Lorentz
• 1
• Catherine Huafei Yan
• 2
1. 1.Department of MathematicsTexas A&M University at QatarDohaQatar
2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA