Analysis of the gift exchange problem
By Moa Apagodu, David Applegate, Neil J. A. Sloane, and Doron Zeilberger
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[Appeared in Electronic J. of Combinatorics, Volume 24(3), paper P3.9]
First Written: Jan. 26, 2017. This version: Feb. 5, 2017.
David Applegate and Neil Sloane, inspired, in large part,
by an insightful comment made by
Bob Proctor regarding OEIS sequence A1515,
wrote a nice earlier version.
In the current version, Moa Apagodu and myself (DZ), joined in, and got
Shalosh B. Ekhad to do some work.
On Line Appendices
Accompanying Maple Programs
Important: This article is accompanied by a the Maple
program
 DavidNeil.txt
that computes the minimalorder recurrences using the (discrete) AlmkvistZeilberger
and minimalorder differential equations for the generating functions.
 EKAHD.txt
A new vesrion of EKHAD.txt that contains procedure Matana(n,K) used below.
Sample Input and Output for DavidNeil

To get the minimal recurrences satisfied by G_{σ}(n), and asymptotics to
order 1/n^{6} for σ from 1 to 6, the
input
yields the
output

To get the minimalorder differential equations satisfied by the generating functions
for G_{σ}(n), g_{σ}(x), for
σ from 1 to 15, the
input
yields the
output
Sample Input and Output for EKHAD.txt