On Promotion and Quasi-Tangled Labelings of Posets

In 2022, Defant and Kravitz introduced extended promotion (denoted $$ \partial ), a map that acts on the set of labelings poset. Extended is generalization Schützenberger’s operator, well-studied permutes linear extensions It known if L labeling an n-element poset P, then ^{n-1}(L) extension. This allows us to regard as sorting operator all where we think P which have been sorted. The requiring n-1 applications be sorted are called tangled; n-2 quasi-tangled. We count quasi-tangled relatively large class posets inflated rooted trees with deflated leaves. Given unique minimal element property has exactly one parent, it follows from aforementioned enumeration this 2(n-1)!-(n-2)! labelings. Using similar methods, outline algorithmic approach enumerating n-k-1 for any fixed k\in \{1,\ldots ,n-2\} . also make partial progress towards proving conjecture maximum possible number tangled