Then there is a single column, which contains files listed with a simple date-time stamp:
20180714112153.jpg
20180714112153.jpg.DS_Store
20180714112153.jpg.DS_Store.DS_Store

This particular script takes advantage of this fact, calculating an approximate number of files in a series of directories, with each directory having approximately the same number of files (or a few less):
#!/bin/bash
#

Reducing powers

Given the function f : {{0, 1,…, n} -> {0, 1,…, n}
where f is a circular shift of the terms. e.g. f((0,1,n,0,1,n,0,1,n)) = (0,1,n,0,1,n,0,1,n)
f(f(f(…f(f(x)))) = f(x)
This function f has a closed form solution. Given N N=…N, if there is a way to reduce the number of power functions needed to find this closed form? N is a natural number.

A:

This form $N!^{ -1}$ is known as the “master function” for the $N$-permutations, and can be easily computed in Wolfram Alpha. For your purpose, all you need is the following identities:
$$(A!)^{ -1}=(A!)^*(A!)^{*2}(A!)^{*4}…$$
$$(A!)^*(A!)^{*k}=\frac{(A!)^{k+1}}{k+1}$$
$$(A!)^n=\frac{n!}{(n-1)!}A^{n-1}$$
Here’s a Mathematica code to calculate, for a given $N$, the number of combinations of $N$ powers, for $N\le 100$ (which may be uncomfortably large):

Cases[{Take[N!^n,{n,1,N}],N!^n,{0,2}]

(* Out[272]= {1,2,{1,2,{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,
17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,
37,38,39,40,41,42,43
0cc13bf012

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F3 3.44 5.68Â January 10, 2018
Date: 15 May 2017For any official information on the course, contact the Australian GovernmentÂ .