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Three-Element Lucky Sets of Stirling Permutations

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Contributed by YURAN XIU ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

We characterize the three-element sets of lucky car indices that occur when a Stirling permutation of order nn is used as a parking function. A set {1,x,y}\{1,x,y\}, with 1<x<y<2n1<x<y<2n, is admissible precisely when xx is odd, or xx is even and yy is odd with x<nx<n, or both are even with x+y≤2nx+y\leq2n. Explicit Stirling words realize every permitted case. The number of admissible sets is (n−1)(n−2)+2nh−3h(h+1)/2(n-1)(n-2)+2nh-3h(h+1)/2, where h=⌊(n−1)/2⌋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 constructions, and enumeration have been checked in Lean. HAL deposit: hal-05800210v1.

References

  1. L. Colmenarejo, A. Dawkins, J. Elder, P. E. Harris, K. J. Harry, S. Kara, D. Smith, and B. E. Tenner, On the lucky and displacement statistics of Stirling permutations , J. Integer Sequences 27 (2024), Article 24.6.7. https://cs.uwaterloo.ca/journals/JIS/VOL27/Tenner/tenner12.pdf.link

Version history

  1. v1Submitted by YURAN XIUInitial depositCurrentOct 18, 2026