The Hook Formula. If $\lambda$ is a partition of $n$ then the number of standard Young tableaux of shape $\lambda$ is
$$f^\lambda = \frac{n!}{\prod_{\alpha \in [\lambda]} h_\alpha} $$
where $h_\alpha$ is the hook-length of the box $\alpha$ in the Young diagram $[\lambda]$ of $\lambda$, as shown below for $(5,4,2,1)$. The special case $\lambda = (n,n)$ gives the Catalan numbers:
$$f^{(n,n)} = C_n = \frac{(2n)!}{(n+1)!n!} = \frac{1}{n+1} \binom{2n}{n}. $$
If $m_k>m_{k-1}>\dots>m_1$ are hook-lengths in the first column of Young diagram $\lambda$, i.e. lengths of rows are $0<m_1\leqslant m_2-1 \leqslant m_3-2\leqslant \dots \leqslant m_k-(k-1)$, then equivalent form is
$$
f^{\lambda}=\frac{n!}{\prod m_i!}\prod_{1\leqslant i<j\leqslant k} (m_j-m_i).
$$
This formula for $f^{\lambda}$ was established by G. Frobenius (Uber die charaktere der symmetrischer gruppe, Preuss. &ad. Wk. sitz. (1900), 516–534.) and A. Young (Quantitative substitutional analysis II, Proc. London Math. Sot., Ser. 1, 35 (1902), 361–397). Equivalence follows from the observation that product of hook lengths in $j$-th row equals $m_j!/\prod_{i<j} (m_j-m_i)$.
The Hook Formula was first proved by Frame, Robinson and Thrall. It is important as a unifying result in enumerative combinatorics. It also gave another early indication (after Nakayama's Conjecture) of the importance of hooks, $p$-cores and $p$-quotients to the representation theory of the symmetric group.
This is just a slight expansion of my comment.
When the environment changes, behavioral parameters (that is, parameters like $M$ that describe people's behavior --- in this case the behavior of public servants deciding how many clients to serve each day) are going to change. Therefore such parameters should not be taken as constants; they should be determined within the model.
This means we need to be able to predict how $M$ will change in an unprecedented circumstance. Fortunately, we have a lot of relevant data. For example, consider the function $f(p)$ that tells you how much income a person is willing to forgo in order to avoid a probability $p$ of death. At least in the United States, we know (somewhat roughly and with various caveats) that when $p$ is small, $f(p)\approx \$10,000,000\times p$. (Theory predicts, and evidence seems to confirm, that $f(p)$ is linear for small $p$.) We infer this, for example, from the premiums you have to pay people in order to get them to take on dangerous jobs, or from the amount people are willing to pay for safety devices. I'm not sure whether $p\approx 1\%$ counts as small for this purpose, but there are data available that will help decide that.
(This, incidentally, is precisely what economists mean when they say that "In the United States, the value of a life is about $10,000,000".)
The best way to account for all this is to assume that people are maximizing some functionn $U$ which takes as arguments things like income, social interaction, time spent being sick, and probability of death. Try to estimate the function $U$ by observing the choices people make in a great variety of ordinary circumstances. In other words, observe their behavioral parameters in ordinary times, assume that those behavioral parameters are the solutions to some maximization problem, and try to infer what's being maximized.
Now when the pandemic comes along, you've got to assume that something is unchanged; otherwise you have no basis to make any predictions whatsoever. The idea is to assume that what's unchanged is the maximand $U$, and that the pandemic represents a change in the constraints subject to which people are trying to maximize. Having estimated $U$, having assumed it's fixed, and writing down the new constraints, you can calculate the new behavioral parameters that result from the new maximization problem.
Now your model is essentially a fixed point problem: Behavioral parameters (that is, the solutions to the maximization problem) cause changes in behavior, which cause changes in the way the pandemic spreads (that is, the constraints on the maximization problem), which cause changes in behavioral parameters. The solution to the model is a fixed point of that process.
You can also estimate how the fixed point will change if you add additional constraints, such as penalties for going outside, prohibitions on meetings, etc.
This sort of modeling is what economists try to do all the time. I expect, but do not know, that one could say the same of epidemiologists. No model is perfect, but economists have learned the painful lesson that some models are a lot less perfect than others, and that fixed behavioral parameters are generally a hallmark of such models.
Best Answer
Complex networks is a young and active area of scientific research inspired largely by the empirical study of real-world networks such as computer networks, social networks, biological networks, ecological networks, telecommunication networks etc. A lot of interesting and important properties of complex networks were found by statistics such as sparsity, small world phenomenon (also called six degree separation, for example, from the ‘small world’ studies of Harvard sociologist Stanley Milgram[1], play of Guare [2], Hands shaking(Watt’s father remarks that he was only six handshakes away from the president of US) example of Watts[3]), power law distribution(also called long tail or heavy tail distribution or scale free, Barabási and Albert [4] noticed actor collaboration networks and World Wide Web had power law distribution).
The field continues to develop at a brisk pace, and has brought together researchers from many areas including mathematics, physics, biology, telecommunications, computer science, sociology, epidemiology, and others[5].
To set up the foundations of complex network, the first step is to define some new metrics. There are a lot of different metrics widely used in complex networks such as average degree, diameter, entropy, average distance, centrality, betweenness, density, clustering coefficient, transitivity, diffusion time etc.
How to model complex networks is one big issue in this area. Traditional well known model such as the popular Erdős-Rényi random network model does not fit on the real world network so well in most cases. For example, although Erdős-Rényi random network has small diameter, but have very few triangles, which is not like in social networks the friends of one person tend to be friends, too. Scientists are trying to find new mathematical models for complex networks.
For small world model, Watts and Strogatz [6] started from ring lattice with $n$ vertices and $k$ edges per vertex , and then rewired each edge with probability $p$, connecting one end to a vertex chosen at random, this can drastically reduce the diameter. Bollobás and Chung [7] added a random matching to a ring of $n$ vertices with nearest neighbor connections and showed the resulting graph had diameter $\sim\log n$. This model is called BC small world, however, this is weak in social network. Newman and Watts [8] (NW small world)added a Poisson number of shortcuts with mean $n\rho/2$ instead of rewiring edges, and then attaches them to randomly chosen pairs of sites. This results in a Poisson mean $\rho$ number of long distance edges per site.
For Power Law distribution model, Barabási and Albert [4] found some real world network such as actor collaboration networks and World Wide Web had degree distributions $p_k\sim Ck^{-\gamma}$ as $k\rightarrow \infty$. Usually $\gamma$ is between $2$ and $3$, but Liljeros et al. [9] found $\gamma_{male}=3.3$ and $\gamma_{female}=3.5$ by analyzing the data of sexual behavior of 4781 Swedes. Ebel et al. [10] found $\gamma=1.81$ by studying email network of Keil University. Newman[11][12][13] found the number of collaborators was better fit by a power law with an exponential cutoff $p_k\sim Ck^{-\tau}exp(-k/k_c)$ in collaboration network. Later, scholars found other values of $\gamma$. Barabási and Albert [4] introduced the preferential attachment model to give a mechanism explanation for power law distribution. Whether this model still keeps small world property? Bollobás and Riordan [14] showed for this model, the diameter $\sim \log n/(\log \log n)$. Chung and Lu[15][16] showed for their corresponding model (known as Chung-Lu model) has the distance between to randomly chosen vertices is $O(\log n/(\log \log n))$. Callaway, Hopcroft, Kleinberg, Newman and Strogatz [17] introduced CHKNS model inspired by Barabási and Albert [4].
Just name a few. Further details and more models are referenced to [18][19].
However, it is difficult to find a model which can fit perfectly on the real world networks in every aspect. Now the research in this area is usually data and application driven. For example, they study application problems such as influence maximization, community detection, rumor detection, recommender system, PageRank etc. , in which they design algorithm or mechanism and use some real world network data to make simulation to test the quality of algorithms, usually they don’t care too much about the model itself. But the big issue without an exact mathematical model is large scale data is hard to handle, thus some sparse optimization and large scale optimization techniques were introduced.
For other application such as the famous Kidney Exchange Long Chain mechanism of Al Roth, who won Nobel Prize, dynamic homogeneous random graph [20], Random compatibility graphs[21], sparse graphs [21]etc. were used. Those models are more mathematical than better fitting on the real world networks.
In communication networks, such as ad hoc sensor network, it is not small world nor power law distribution, random geometry network or other models are used.
[1] S. Milgram, “The small world problem,” Psychology Today, vol. 2, no. 1, pp.60–67, 1967.
[2] J. Guare, Six Degrees of Separation: A Play. New York: Vintage, 1990.
[3] Watts, Duncan J. Six degrees: The science of a connected age. WW Norton & Company, 2004.
[4] Barabási, Albert-László, and Réka Albert. "Emergence of scaling in random networks." science 286.5439 (1999): 509-512.
[5] A.E. Motter, R. Albert (2012). "Networks in Motion". Physics Today 65 (4): 43–48.
[6] Watts, Duncan J., and Steven H. Strogatz. "Collective dynamics of ‘small-world’networks." nature 393.6684 (1998): 440-442.
[7]Bollobás, Bela, and Fan R. K. Chung. "The diameter of a cycle plus a random matching." SIAM Journal on discrete mathematics 1.3 (1988): 328-333.
[8] Newman, Mark EJ, and Duncan J. Watts. "Renormalization group analysis of the small-world network model." Physics Letters A 263.4 (1999): 341-346.
[9]Liljeros, Fredrik, et al. "The web of human sexual contacts." Nature 411.6840 (2001): 907-908.
[10] Ebel, Holger, Lutz-Ingo Mielsch, and Stefan Bornholdt. "Scale-free topology of e-mail networks." Physical review E 66.3 (2002): 035103.
[11] Newman, Mark EJ. "The structure of scientific collaboration networks." Proceedings of the National Academy of Sciences 98.2 (2001): 404-409.
[12] Newman, Mark EJ. "Scientific collaboration networks. I. Network construction and fundamental results." Physical review E 64.1 (2001): 016131.
[13] Newman, Mark EJ. "Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality." Physical review E 64.1 (2001): 016132.
[14] Bollobás, Béla, and Oliver Riordan. "The diameter of a scale-free random graph." Combinatorica 24.1 (2004): 5-34.
[15] Chung, Fan, and Linyuan Lu. "Connected components in random graphs with given expected degree sequences." Annals of combinatorics 6.2 (2002): 125-145.
[16] Chung, Fan, and Linyuan Lu. "The average distance in a random graph with given expected degrees." Internet Mathematics 1.1 (2004): 91-113.
[17] Callaway, D. S., Hopcroft, J. E., Kleinberg, J. M., Newman, M. E., & Strogatz, S. H. (2001). Are randomly grown graphs really random?. Physical Review E, 64(4), 041902.
[18] Chung, Fan RK, and Linyuan Lu. Complex graphs and networks. Vol. 107. Providence: American mathematical society, 2006.
[19] Durrett, Richard. Random graph dynamics. Vol. 200. No. 7. Cambridge: Cambridge university press, 2007.
[20] Ashlagi, Itai, Patrick Jaillet, and Vahideh H. Manshadi. "Kidney exchange in dynamic sparse heterogenous pools." arXiv preprint arXiv:1301.3509 (2013).
[21] Ashlagi, Itai, et al. The need for (long) chains in kidney exchange. No. w18202. National Bureau of Economic Research, 2012.