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MATH318 Combinatorics课程简介
Applied mathematics is the mathematical study of general scientific concepts, principles, and phenomena that, because of their widespread occurrence and application, relate or unify various disciplines. The core of the program at MIT concerns the following principles and their mathematical formulations: propagation, equilibrium, stability, optimization, computation, statistics, and random processes.
Sophomores interested in applied mathematics typically survey the field by enrolling in 18.200 and 18.300 Principles of Applied Mathematics. Subject 18.200 is devoted to the discrete aspects of the study and may be taken concurrently with 18.03. It carries CI-M credit in mathematics. Subject 18.300, given only in the second term, is devoted to continuous aspects and makes considerable use of differential equations.
The subjects in Group I of the program correspond roughly to those areas of applied mathematics that make heavy use of discrete mathematics, while Group II emphasizes those subjects that deal mainly with continuous processes. Some subjects, such as probability or numerical analysis, have both discrete and continuous aspects.
Students planning to go on to graduate work in applied mathematics should also take some basic subjects in analysis and algebra.
PREREQUISITES
Combinatorics is a branch of mathematics with broad areas of application. There are important uses of combinatorics in computer science, operations research, probability, and statistics. Theoretical thermodynamics uses combinatorial theory to describe ideas such as entropy. The combinatorial analysis is a cornerstone of the study of error-correcting codes; these codes are used to transmit information from deep space or to protect the quality of music on compact discs. Our course will mainly focus on describing and/or counting complicated sets. Often questions that begin “How many ways can you…?” or “How many steps does it take to…?” are answered using combinatorial analysis. Such questions on the surface may appear rather uninteresting, but one can quickly get to questions that are quite engaging. What gambler wouldn’t want to understand the odds at winning a poker hand?
MATH318 Combinatorics HELP（EXAM HELP， ONLINE TUTOR）
Let ten balls be given. How many ways are there to put these into four boxes such that three boxes have capacity three and one has capacity four? How many possibilities are there if we suppose that from the ten balls seven is blue, three is red, and the red balls cannot share a box?
Show that the $r$-Eulerian numbers have the following special values
$$
\begin{aligned}
\left\langle\begin{array}{c}
n \
0
\end{array}\right\rangle_r & =r ! \
\left\langle\begin{array}{l}
n \
n
\end{array}\right\rangle_r & =r ! r^n \
\left\langle\begin{array}{c}
n \
n-1
\end{array}\right\rangle_r & =r !\left[(r+1)\left((r+1)^n-r^n\right)-n r^n\right] .
\end{aligned}
$$
Prove that the generating function of the hyperharmonic numbers is
$$
\sum_{n=0}^{\infty} H_n^r x^n=-\frac{\ln (1-x)}{(1-x)^r} .
$$
(The exponential generating function is more complicated, see [181, 185] for the details with respect to the ordinary harmonic numbers and $[425,421]$ for the exponential generating function of the hyperharmonic numbers.)
Based on Section 7.2 , prove the inequality
$$
\frac{(k+r)^n}{k !}-\frac{(k-1+r)^n}{(k-1) !}<\left{\begin{array}{l} n+r \ k+r \end{array}\right}_r<\frac{(k+r)^n}{k !} $$ for all $n \geq m>0$.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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