A Short Proof that the number of (a,b)-parking functions of length n is a(a+bn)n-1

By AJ Bu and Doron Zeilberger

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Originally Written: Dec. 15, 2024

This version: Dec. 19, 2024 (adding feedback from Richard Stanley)

We give a very short proof of the fact that the number of (a,b)-parking functions of length n equals a times (a+bn) to the power (n-1). This was first proved in 2003 by Kung and Yan, via a very long and torturous route, as a corollary of a more general result. This new version contains a reference to previous work kindly communicated by Richard Stanley.

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