Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
18
The categories of this paper are: econ
19
The categories of this paper are: eess
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin