Three-Element Lucky Sets of Stirling Permutations
We characterize the three-element sets of lucky car indices that occur when a Stirling permutation of order $n$ is used as a parking function. A set $\{1,x,y\}$, with $1<x<y<2n$, is admissible precisely when $x$ is odd, or $x$ is even and $y$ is odd with $x<n$, or both are even with $x+y\leq2n$. Explicit Stirling words realize every permitted case. The number of admissible sets is $(n-1)(n-2)+2nh-3h(h+1)/2$, where $h=\lfloor(n-1)/2\rfloor$. This answers the three-element characterization and enumeration question in Problem 46 of Colmenarejo et al. The complete classification, parking…
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