# Loop with variable nesting depth and variable count at each level

• Mathematica
• Swamp Thing
Pythondef factorial(n): if n<=0: return 1 elif n==1: return 1 elif n<=2: return 1*factorial(n-1) elif n<=3: return 2*factorial(n-1) elif n<=4: return 3*factorial(n-1) elif n<=5: return 4*factorial(

#### Swamp Thing

TL;DR Summary
"M" levels of nested loops;
each loop running from 1 to Counts[[m]]
In Mathematica, how can we have "M" levels of nested loops, with each loop (level) going from 1 to Counts=##\{N_1, N_2, ... N_M\}## respectively, where the value of M and the elements of Counts are computed beforehand?

Swamp Thing said:
TL;DR Summary: "M" levels of nested loops;
each loop running from 1 to Counts[[m]]

In Mathematica, how can we have "M" levels of nested loops, with each loop (level) going from 1 to Counts=##\{N_1, N_2, ... N_M\}## respectively, where the value of M and the elements of Counts are computed beforehand?

Do a single loop of length $N_1N_2 \dots N_M$ and calculate the values of the separate indices $(i_1, \dots, i_M)$ in each iteration based on the single index $$1 \leq i = 1 + (i_1 - 1) + N_1(i_2 - 1) + N_1N_2(i_3 - 1) + \dots + (N_1N_2 \cdots N_{M-1})(i_M - 1) \leq (N_1 \cdots N_{M})$$ as $$\begin{split} i_1 &= 1 + (i \mod N_1) \\ i_2 &= 1 + ((i - i_1)/N_1 \mod N_2) \\ i_3 &= 1 + ((i - i_1 - N_1i_2)/N_2 \mod N_3) \end{split}$$ etc.

EDIT: More efficient is to increment $i_1$ after each loop, and increment $i_2$ and reset $i_1$ if this makes $i_1 > N_1$, etc.

Last edited:
jedishrfu said:
There is a discussion on stackexchange about using recursion
pasmith said:
Do a single loop of length N1N2…NM and calculate the values of the separate indices

Thanks, I will do both methods as an exercise to keep from getting rusty. I have always felt a bit daunted by recursive code, so this may be a good time to lay that to rest. And pasmith's method seems to have a lot of moving parts, which will be interesting to master.

Code:
// Initialize loop counters
for i = 1 to M:
I[i] = 1

while true:

// do stuff

// Increment I
I += 1

for i = 1  to M-1:
// If I[i] has advanced beyond Counts[i], set it to 1 and increment I[i+1].
if I[i] > Counts[i]:
I[i] = 1
I[i+1] += 1

// As soon as I[M] is out of range we are done.
if I[M] > Counts[M]:
break

• Swamp Thing
Here's am example of factorial in Mathematica from the RosettaCode site:

## Mathematica / Wolfram Language​

Note that Mathematica already comes with a factorial function, which can be used as e.g. 5! (gives 120). So the following implementations are only of pedagogical value.

### Recursive​

Code:
factorial[n_Integer] := n*factorial[n-1]
factorial = 1

### Iterative (direct loop)​

Code:
factorial[n_Integer] :=
Block[{i, result = 1}, For[i = 1, i <= n, ++i, result *= i]; result]

### Iterative (list)​

Code:
factorial[n_Integer] := Block[{i}, Times @@ Table[i, {i, n}]]

https://rosettacode.org/wiki/Factorial#Mathematica_/_Wolfram_Language

and more languages implementing factorial:

https://rosettacode.org/wiki/Factorial

I like this site because of its numerous examples and language comparisons.

With respect to recursion, the recursing method has a conditional that stops the recursion at the start of the method, which is best illustrated in Python:

Recursive
Python:
def factorial(n):

if n<1:      ## stops recursion and blocks negative n values
return 1

return n*factorial(n-1)

Last edited:
• Swamp Thing

## 1. What is a loop with variable nesting depth and variable count at each level?

A loop with variable nesting depth and variable count at each level is a programming structure that allows for the repetition of a code block with a varying number of levels and iterations at each level. This means that the loop can be nested within itself multiple times, and the number of iterations executed at each level can change depending on the conditions set by the programmer.

## 2. How does a loop with variable nesting depth and variable count at each level differ from a regular loop?

A regular loop has a fixed number of levels and a fixed number of iterations at each level. In contrast, a loop with variable nesting depth and variable count at each level allows for more flexibility and adaptability in terms of the structure and number of iterations. This can be useful when dealing with complex or changing data structures.

## 3. What are some common use cases for a loop with variable nesting depth and variable count at each level?

This type of loop is often used in data processing and manipulation tasks, especially when dealing with nested data structures. It can also be useful in situations where the number of iterations at each level may vary, such as in a game with multiple levels or in simulations where the number of iterations needed may change based on user input or other factors.

## 4. How is the nesting depth and count of a loop determined?

The nesting depth and count of a loop are determined by the conditions set by the programmer. These conditions can be based on user input, data values, or other factors. The loop will continue to execute until the conditions are met or until a break statement is encountered.

## 5. What are some potential challenges when using a loop with variable nesting depth and variable count at each level?

One potential challenge is keeping track of the nested levels and iterations, which can become complex and difficult to debug if not done carefully. Additionally, if the conditions for the loop are not carefully considered, it may result in an infinite loop, which can cause the program to crash or become unresponsive.