Promotion, Tangled Labelings, and Sorting Generating Functions
Margaret Bayer, Herman Chau, Mark Denker, Owen Goff, Jamie Kimble, Yi-Lin Lee, and Jinting Liang

TL;DR
This paper explores promotion operators on labelings of finite posets, proving conjectures for specific poset classes and introducing generating functions to analyze labeling transformations.
Contribution
It strengthens existing conjectures for certain poset classes and introduces new generating functions to study promotion dynamics.
Findings
Proved the stronger tangled labeling conjecture for inflated rooted forest and shoelace posets.
Established log-concavity of coefficients in the cumulative generating function for ordinal sums of antichains.
Refined the weak order on the symmetric group through new generating function analysis.
Abstract
We study Defant and Kravitz's generalization of Sch\"utzenberger's promotion operator to arbitrary labelings of finite posets in two directions. Defant and Kravitz showed that applying the promotion operator times to a labeling of a poset on elements always gives a natural labeling of the poset and called a labeling tangled if it requires the full promotions to reach a natural labeling. They also conjectured that there are at most tangled labelings for any poset on elements. In the first direction, we propose a further strengthening of their conjecture by partitioning tangled labelings according to the element labeled and prove that this stronger conjecture holds for inflated rooted forest posets and a new class of posets called shoelace posets. In the second direction, we introduce sorting generating functions and cumulative generating functions for…
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