Generating functions for Wilf equivalence under generalized factor order

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set \((P, \leq_P)\) by setting \(u \leq_P w\) if there is a subword \(v\) of \(w\) of the same length as \(u\) such that the \(i\)-th character of \(v\) is greater than or equal to the \(i\)-th character of \(u\) for all \(i\). This subword \(v\) is called an embedding of \(u\) into \(w\). For the case where \(P\) is the positive integers with the usual ordering, they defined the weight of a word \(w = w_1\ldots w_n\) to be \(\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}\), and the corresponding weight generating function \(F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)\). They then defined two words \(u\) and \(v\) to be Wilf equivalent, denoted \(u \backsim v\), if and only if \(F(u;t,x) = F(v;t,x)\). They also defined the related generating function \(S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)\) where \(\mathcal{S}(u)\) is the set of all words \(w\) such that the only embedding of \(u\) into \(w\) is a suffix of \(w\), and showed that \(u \backsim v\) if and only if \(S(u;t,x) = S(v;t,x)\). We continue this study by giving an explicit formula for \(S(u;t,x)\) if \(u\) factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if \(u \backsim v\) then \(u\) and \(v\) must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection \(f: \mathcal{S}(u) \rightarrow \mathcal{S}(v)\) such that \(f(u)\) is a rearrangement of \(u\) for all \(u\).