# 组合学 Combinatorics MATH344

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look at the complete words. Thus,
$$\operatorname{Pr}\left(X_{t}=h \mid \varkappa=t\right)=\frac{1}{N} \cdot \sum_{\omega} \operatorname{Pr}\left(\omega^{\bar{\alpha}}=h\right)$$
where we sum over the complete words $\omega$ of length $t$ such that no shorter word is a complete word. Therefore from Theorem 1, $\omega$ is uniform. So,
$$\operatorname{Pr}\left(X_{t}=h \mid \varkappa=t\right)=\frac{1}{N} \cdot N \cdot \frac{1}{|G|}=\frac{1}{|G|}$$
Thus, $\varkappa$ is strong uniform.
Note, we can generalise this to any nilpotent group with generators corresponding to our generators, and the mixing time $=O(|\Lambda| \log |\Lambda|)$.

## ECON344COURSE NOTES ：

Proof: Let $y \in G$, and write $y$ in the form . We can write
$$\varphi(x)-\varphi(x y)=\left[\varphi(x s)-\varphi\left(x s_{1}\right)\right]+\ldots+\left[\varphi\left(x s_{1} \ldots s_{\ell-1}-\varphi(x y)\right]\right.$$
It follows, for example, by the Cauchy-Schwarz inequality that
$$(\varphi(x)-\varphi(x y))^{2} \leq \ell^{} \sum_{i=1}^{\ell}\left(\varphi\left(x s_{1} \ldots s_{i-1}\right)-\varphi\left(x s_{1} \ldots s_{i}\right)\right)^{2}$$ where $\ell^{}$ is the number of nonzero terms in the sum, and is bounded above by $d=\operatorname{diam}(\bar{C})$, since $\gamma$ is geodesic. Summing this inequality over $x \in G$ we get
$$\sum_{x \in G}(\varphi(x)-\varphi(x y))^{2} \leq d \sum_{z \in G, s \in S} N_{\gamma}(s, \bar{C})(\varphi(z)-\varphi(z s))^{2}$$

# 组合学|MTH3021RWA Combinatorics代写

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The definition of boxed products implies the coefficient relation
$$A_{n}=\sum_{k=1}^{n}\left(\begin{array}{l} n-1 \ k-1 \end{array}\right) B_{k} C_{n-k} .$$
The binomial coefficient that appears in the standard labelled product is now modified since only $n-1$ labels need to be distributed between the two components, $k-1$ going to the $\mathcal{B}$ component (that is constrained to contain the label 1 already) and $n-k$ to the $\mathcal{C}$ component. From the equivalent form
$$\frac{A_{n}}{n !}=\frac{1}{n} \sum_{k=0}^{n}\left(\begin{array}{l} n \ k \end{array}\right)\left(k B_{k}\right) C_{n-k}$$

## MTH3021RWA COURSE NOTES ：

A useful special case is the min-rooting operation,
$$\mathcal{A}={1}^{0} \star \mathcal{C} \text {. }$$
for which a variant definition goes as follows. Take in all possible ways elements $\gamma \in \mathcal{C}$, prepend an atom with a label smaller than the labels of $\gamma$, for instance 0 , and relabel in the canonical way over $[1 \ldots(n+1)]$ by shifting all label values by 1 . Clearly $A_{n+1}=C_{n}$ which yields
$$A(z)=\int_{0}^{z} C(t) d t$$
a result also consistent with the general formula of boxed products.
For some applications, it is easier to impose constraints on the maximal label rather than the minimum. The max-boxed product written
$$A=\left(B^{-} * \mathcal{C}\right) \text {. }$$

# 组合学 Combinatorics MATH314301

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Let $y \in G$, and write $y$ in the form (1). We can write
$$\varphi(x)-\varphi(x y)=\left[\varphi(x 8)-\varphi\left(x s_{1}\right)\right]+\ldots+\left[\varphi\left(x s_{1} \ldots s_{\ell-1}-\varphi(x y)\right] .\right.$$
It follows, for example, by the Canchy-Schwarz inequality that
$$(\varphi(x)-\varphi(x y))^{2} \leq \ell^{*} \sum^{\ell}\left(\varphi\left(x s_{1} \ldots s_{i-1}\right)-\varphi\left(x s_{1} \ldots s_{i}\right)\right)^{2}$$

where $\ell^{*}$ is the number of nonzero terms in the sum, and is bounded above by $d=\operatorname{diam}(\bar{C})$, since $\gamma$ is geodesic. Summing this inequality over $x \in G$ we get
$$\sum_{x \in G}(\varphi(x)-\varphi(x y))^{2} \leq d \sum_{t \in C, s \in S} N_{\gamma}(s, \bar{C})(\varphi(z)-\varphi(z s))^{2}$$
Since this holds for all $y \in G$, we may average the left hand side with respect to $y$ with weights $\approx(y)$ to get
$$\sum_{x, y \in C}(\varphi(x)-\varphi(x y))^{2} \tilde{\pi}(y) \leq d \sum_{z \in G, s \in S} N_{\gamma}(s, \bar{C})(\varphi(z)-\varphi(z s))^{2}$$

## MATH314301COURSE NOTES ：

Last time, we proved
Theorem 1 Let $C$ be a subset of the group $G, S=S^{-1}$ a symmetric generating set, $\pi=U(S)$ the uniform distribution on $S$, and $p_{\pi}$ the “one-step evolution” of the random walk (i.e. $p_{\pi} \varphi=U(S) * \varphi$ ). Then for any probability distribution $\varphi$,
$$\left|p_{n} \varphi\right|^{2} \leq\left(1-\frac{|G \backslash C|}{2 \cdot A \cdot|G|}\right) \cdot|\varphi|^{2}$$
where $A=d \cdot|S| \cdot \max {s \in S} \max {g \in C} \mu_{s}(g), d=\operatorname{diam}(\bar{C}, G)$.
We will use this theorem to bound the escape time of a random walk $X_{\mathrm{t}}$ generated by $S$. For a subset $C$ of $G$, set
$$\varphi_{t}(g)=\operatorname{Pr}\left[X_{t}=g \text { and } X_{i} \in C \forall i=1 \ldots t\right]$$
Obviously, supp $\varphi_{t} \subset C,\left|\varphi_{0}\right| \leq 1$ ( 1 if $C$ contains $1_{G}, 0$ otherwise) and