In how many distinct ways can flocks be formed? A problem in sheep combinatorics

In this short paper, we extend the concept of the strict order polynomial \(\Omega_{P}^{\circ}(n)\), which enumerates the number of strict order-preserving maps \(\phi:P\rightarrow\boldsymbol{n}\) for a poset \(P\), to the extended strict order polynomial \(\text{E}_{P}^{\circ}(n,z)\), which enumerates analogous maps for the elements of the power set \(\mathcal{P}(P)\). The problem at hand immediately reduces to the problem of enumeration of linear extensions for the subposets of \(P\). We show that for every \(Q\subset P\) a given linear extension \(v\) of \(Q\) can be associated with a unique linear extension \(w\) of \(P\). The number of such linear extensions \(v\) (of length \(k\)) associated with a given linear extension \(w\) of \(P\) can be expressed compactly as \(\binom{\text{del}_{P}(w)}{k}\), where \(\text{del}_{P}(w)\) is the number of deletable elements of \(w\) defined in the text. Consequently the extended strict order polynomial \(\text{E}_{P}^{\circ}(n,z)\) can be represented as \( \text{E}_{P}^{\circ}(n,z)=\sum_{w\in\mathcal{L}(P)}\sum_{k=0}^{p}\binom{\text{del}_{P}(w)}{p-k}\binom{n+\text{des}(w)}{k}z^{k}\). The derived equation can be used for example for solving the following combinatorial problem: Consider a community of \(p\) shepherds, some of whom are connected by a master-apprentice relation (expressed as a poset \(P\)). Every morning, \(k\) of the shepherds go out and each of them herds a flock of sheep. Community tradition stipulates that each of these \(k\) shepherds will herd at least one and at most \(n\) sheep, and an apprentice will always herd fewer sheep than his master (or his master's master, etc). In how many ways can the flocks be formed? The strict order polynomial answers this question for the case in which all \(p\) shepherds go to work, and the extended strict order polynomial considers also all the situations in which some of the shepherds decide to take a day off.