Calculating Complexity with Big-O Notation

[Hurkyl]

$$\frac{<N!>}{<(N-n)!>} = <N^n> (1 + \mathcal{O}( \frac{1}{<N!>} ))$$

The popup you get from clicking on this is missing lots of the code. I presume because my browser is trying to render things as HTML tags.

 

[franznietzsche]

it is missing every instance of

$$<N!>$$

Weird.

And it is on the popup for that image. Seems to be just that one part gets dropped.

 

[robphy]

$$\begin{align*}<br /> <N!>& <N! > < N!> < N! > < N ! > <N> <N > < N> < N > <!> - \\<br /> <P!>& <P! > < P!> < P! > < P ! > <P> <P > < P> < P > <!> - \\<br /> <Q!>& <Q! > < Q!> < Q! > < Q ! > <Q> <Q > < Q> < Q > <!> - \\<br /> <I!>& <I! > < I!> < I! > < I ! > <I> <I > < I> < I > <!> - \\<br /> <U!>& <U! > < U!> < U! > < U ! > <U> <U > < U> < U > <!> - \\<br /> <B!>& <B! > < B!> < B! > < B ! > <B> <B > < B> < B > <!> <br /> \end{align*}$$

$$<\psi>$$

$$<\psi | H |\psi>$$

\begin{align*}
<N!>& <N! > < N!> < N! > < N ! > <N> <N > < N> < N > <!> - \\
<P!>& <P! > < P!> < P! > < P ! > <P> <P > < P> < P > <!> - \\
<Q!>& <Q! > < Q!> < Q! > < Q ! > <Q> <Q > < Q> < Q > <!> - \\
<I!>& <I! > < I!> < I! > < I ! > <I> <I > < I> < I > <!> - \\
<U!>& <U! > < U!> < U! > < U ! > <U> <U > < U> < U > <!> - \\
<B!>& <B! > < B!> < B! > < B ! > <B> <B > < B> < B > <!> 
\end{align*}