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Promotion Sorting

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Abstract 

Schützenberger’s promotion operator is an extensively-studied bijection that permutes the linear extensions of a finite poset. We introduce a natural extension ∂ of this operator that acts on all labelings of a poset. We prove several properties of ∂; in particular, we show that for every labeling L of an n-element poset P, the labeling ∂n− 1(L) is a linear extension of P. Thus, we can view the dynamical system defined by ∂ as a sorting procedure that sorts labelings into linear extensions. For all 0 ≤ k ≤ n − 1, we characterize the n-element posets P that admit labelings that require at least n − k − 1 iterations of ∂ in order to become linear extensions. The case in which k = 0 concerns labelings that require the maximum possible number of iterations in order to be sorted; we call these labelings tangled. We explicitly enumerate tangled labelings for a large class of posets that we call inflated rooted forest posets. For an arbitrary finite poset, we show how to enumerate the sortable labelings, which are the labelings L such that ∂(L) is a linear extension.

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Data Availability Statement

Data availability is not applicable to this paper since we did not use any data.

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Acknowledgements

We are grateful to James Propp for suggesting the idea to define and investigating an extension of promotion. We also thank Brice Huang for engaging in helpful discussions about previous work on promotion. We thank Darij Grinberg for several helpful comments. The first author was supported by a Fannie and John Hertz Foundation Fellowship and an NSF Graduate Research Fellowship. We thank the anonymous referees for several suggestions that improved the quality of the paper.

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  1. Fine Hall, 304 Washington Rd., Princeton, NJ, 08544, USA

    Colin Defant

  2. Grace Hopper College, Zoom University at Yale, New Haven, CT, 06510, USA

    Noah Kravitz

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  1. Colin Defant
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  2. Noah Kravitz
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Correspondence to Noah Kravitz.

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Defant, C., Kravitz, N. Promotion Sorting. Order 40, 199–216 (2023). https://doi.org/10.1007/s11083-022-09603-9

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