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 Sch\"{u}tzenberger'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. In addition to computing the sizes of the fibers of promotion for rooted tree posets, 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.