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