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You are here: Homepage » Seminars - Academic year 2023-24 » Combinatorics Seminar » Joseph P.S. Kung

seminars:comb:abstract.200112kung

Let **u** be a sequence of non-decreasing positive integers. A **u**-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 Goncharov polynomials g_{n}(x; a_{0},a_{1},…,a_{n-1}) are polynomials defined by the biorthogonality relation:

epsilon(a_{i}) D^{i} g_{n}(x; a_{0},a_{1},…,a_{n-1}) = n! delta_{in} ,

where epsilon(a) is evaluation at a. Goncharov polynomials form a ``natural basis` of polynomials for working with `

**u**-parking functions. For example, the number of **u**-parking functions of length n is (-1)^{n} g_{n}(0; u_{1},u_{2}, …,u_{n}).
Goncharov polynomials also satisfy a linear recursion obtained by expanding x^{n} as a linear combination of Goncharov polynomials. The combinatorial structure underlying this recursion is a decomposition of an arbitrary sequence of positive integers into two subsequences: a ``maximum**u**-parking function and a subsequence consisting of terms of higher values. From this combinatorial decomposition, we derive linear recursions for sum enumerators, expected sums of **u**-parking functions, and higher moments of sums of **u**-parking functions. These recursions yield explicit formulas for these quantities in terms of Goncharov polynomials.

This is joint work with Catherine Yan.

seminars/comb/abstract.200112kung.txt · Last modified: 2020/01/29 14:03 (external edit)

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