Penney's game for permutations

Sergi Elizalde; Yixin Lin

doi:10.46298/dmtcs.16361

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arXiv · 2404.06585

Penney's game for permutations

Sergi Elizalde·Yixin Lin

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Abstract

We consider the permutation analogue of Penney's game for words. Two players, in order, each choose a permutation of length $k\ge3$; then a sequence of independent random values from a continuous distribution is generated, until the relative order of the last $k$ numbers coincides with one of the chosen permutations, making that player the winner. We compute the winning probabilities for all pairs of permutations of length 3 and some pairs of length 4, showing that, as in the original version for words, the game is non-transitive. Our proofs introduce new bijections for consecutive patterns in permutations. We also give some formulas to compute the winning probabilities more generally, and conjecture a winning strategy for the second player when $k$ is arbitrary.

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Sergi Elizalde, Yixin Lin. 2026-07-06. Penney's game for permutations. https://doi.org/10.46298/dmtcs.16361

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