oliadas73

2022-09-03

Let $k\ge 1$ be fix and ${b}_{n}$ be the amount of possible words $w={v}_{1}\cdots {v}_{n}$ of length $n$ on the alphabet $\{1,\dots ,k\}$, such that ${v}_{i}\ne {v}_{i+1},\phantom{\rule{thickmathspace}{0ex}}1\le i\le n-1$.

a) Show by counting that

$${b}_{0}=1\text{and}{b}_{n}=k(k-1{)}^{n-1}\text{for}n\ge 1.$$.

b) Identify the generating function $\sum _{n\ge 0}{b}_{n}{x}^{n}$

My try:

a) first. For the first element of each word there are $k$ possibilities. For every successor there are $$(k-1)$$ possibilities because they depend on the element before themselves.

Is this correct and complete?

a) Show by counting that

$${b}_{0}=1\text{and}{b}_{n}=k(k-1{)}^{n-1}\text{for}n\ge 1.$$.

b) Identify the generating function $\sum _{n\ge 0}{b}_{n}{x}^{n}$

My try:

a) first. For the first element of each word there are $k$ possibilities. For every successor there are $$(k-1)$$ possibilities because they depend on the element before themselves.

Is this correct and complete?

Kaleb Harrell

Beginner2022-09-04Added 14 answers

Yes, your proof for a) is correct. For the generating function, observe that

$$\sum _{n=0}^{\mathrm{\infty}}{b}_{n}{x}^{n}=1+kx\sum _{n=0}^{\mathrm{\infty}}[(k-1)x{]}^{n}$$

and recall what a geometric series is.

$$\sum _{n=0}^{\mathrm{\infty}}{b}_{n}{x}^{n}=1+kx\sum _{n=0}^{\mathrm{\infty}}[(k-1)x{]}^{n}$$

and recall what a geometric series is.

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