Expected Sums of General Parking Functions

Abstract

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.

Parking functions, multi-shuffle, and asymptotic phenomena

“…, which form a natural basis for working with u-parking functions, as well as Abel's multinomial theorem. In particular, our new perspective on parking functions leads to asymptotic moment calculations for multiple coordinates of (a, b)-parking functions that complement the work of Kung and Yan [10], where the explicit formulas for the first and second factorial moments and a general form for the higher factorial moments of sums of (a, b)-parking functions were given.…”

“…, and (1,2). Then (2, 7, 2, 9, 10, 1) ∈ MS(6, 8) is a multi-shuffle of the three words (2, 1, 2), (7), and (10,9).…”

An Experimental Mathematics Approach to the Area Statistic of Parking Functions

“…Theorem 1. (equivalent to a result in [KY1]): The expectation of the area statistic on parking functions of length n is…”

“…Theorem 1. (equivalent to a result in [KY1]): The expectation of the area statistic on parking functions of length n is E 1 (n) := − n 2 + 1 2 (n + 1)! (n + 1) n n−1 k=0 (n + 1) k k!…”

Parking functions: Interdisciplinary connections

“…Knowledge on PF(m, n) with specified parking preferences of l ≤ m cars therefore adds to the understanding of u-parking functions as well. In particular, our enumeration of parking completions provides a different perspective on the volume formula for Pitman-Stanley polytopes [15], and our mixed moment calculations for multiple coordinates of parking functions extend that of Kung and Yan [13], where the explicit formulas for the first and second factorial moments and a general form for the higher factorial moments of sums of u-parking functions were given.…”