Gonc̆arov polynomials and parking functions

Kung, Joseph P. S.; Yan, Catherine H.

doi:10.1016/s0097-3165(03)00009-8

Cited by 48 publications

(64 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The univariate version was discussed in [11] with the differentiation operator and in [13] with an arbitrary delta operator. Here we extend the theory to many variables.…”

Section: Delta Gončarov Polynomialsmentioning

confidence: 99%

“…It is well known that certain values of (univariate) Gončarov polynomials are connected with order statistics (e.g., see [9]). This connection has been further developed [11] into a complete correspondence between Gončarov polynomials and parking functions, a discrete structure lying at the heart of combinatorics. In [10], difference Gončarov polynomials were studied.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Multivariate delta Gončarov and Abel polynomials

Lorentz

Tringali

Yan

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Classical Gončarov polynomials are polynomials which interpolate derivatives. Delta Gončarov polynomials are polynomials which interpolate delta operators, e.g., forward and backward difference operators. We extend fundamental aspects of the theory of classical bivariate Gončarov polynomials and univariate delta Gončarov polynomials to the multivariate setting using umbral calculus. After introducing systems of delta operators, we define multivariate delta Gončarov polynomials, show that the associated interpolation problem is always solvable, and derive a generating function (an Appell relation) for them. We show that systems of delta Gončarov polynomials on an interpolation grid Z ⊆ R d are of binomial type if and only if Z = AN d for some d × d matrix A. This motivates our definition of delta Abel polynomials to be exactly those delta Gončarov polynomials which are based on such a grid. Finally, compact formulas for delta Abel polynomials in all dimensions are given for separable systems of delta operators. This recovers a former result for classical bivariate Abel polynomials and extends previous partial results for classical trivariate Abel polynomials to all dimensions.

show abstract

“…The univariate version was discussed in [11] with the differentiation operator and in [13] with an arbitrary delta operator. Here we extend the theory to many variables.…”

Section: Delta Gončarov Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multivariate delta Gončarov and Abel polynomials

Lorentz

Tringali

Yan

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…The proof begins with a combinatorial decomposition which first appeared in the enumeration of parking functions [6,9]. This decomposition yields an Appell relation, an "umbral" generating function with the power z n replaced by a power series z n φ n (z).…”

Section: Paths In the Planementioning

confidence: 99%

“…In this section, we briefly describe analogs of our results for parking functions. We shall freely use results from [9].…”

Section: Parking Functionsmentioning

confidence: 99%

Lattice and Schröder paths with periodic boundaries

Kung

Mier

Sun

et al. 2009

Journal of Statistical Planning and Inference

Self Cite

View full text Add to dashboard Cite

“…This paper is a companion to our earlier papers [6,7] in which we derive recursions and formulas for moments of sums of parking functions. We shall freely use notations and results from [6].…”

Section: Introductionmentioning

confidence: 99%

Expected Sums of General Parking Functions

Kung

Yan

2003

Annals of Combinatorics

Self Cite

View full text Add to dashboard Cite

A (u 1 , u 2 ,...)-parking function of length n is a sequence (x 1 , x 2 ,..., x n ) whose order statistics (the sequence (x (1) , x (2) ,..., x (n) ) obtained by rearranging the original sequence in non-decreasing order) satisfy x (i) ≤ u i . The Goncarov polynomials g n (x; a 0 , a 1 ,..., a n−1 ) are polynomials biorthogonal to the linear functionals ε(a i )D i , where ε(a) is evaluation at a and D is differentiation. In this paper, we give explicit formulas for the first and second moments of sums of u-parking functions using Goncarov polynomials by solving a linear recursion based on a decomposition of the set of sequences of positive integers. We also give a combinatorial proof of one of the formulas for the expected sum. We specialize these formulas to the classical case when u i = a + (i − 1)b and obtain, by transformations with Abel identities, different but equivalent formulas for expected sums. These formulas are used to verify the classical case of the conjecture that the expected sums are increasing functions of the gaps u i+1 − u i . Finally, we give analogues of our results for real-valued parking functions.

show abstract

Gonc̆arov polynomials and parking functions

Cited by 48 publications

References 19 publications

Multivariate delta Gončarov and Abel polynomials

Multivariate delta Gončarov and Abel polynomials

Lattice and Schröder paths with periodic boundaries

Expected Sums of General Parking Functions

Contact Info

Product

Resources

About