I need to find all integer solutions of these multivariable equations (and many more similar equations)

I have the boundary conditions that b-1 ≥ a1 > a2 ≥ 0

How can I do this in Maple?

For context, b is a base and a1 and a2 are digits in a b-adic number where a1 is bigger than a2. Solutions to these equations correspond to numbers that have a particular property.

I have posted this question on math.stackexchange, but saught an algebraic approach rather than a computational one. Here it is shown that a1 and a2 are linear functions of b. This is where the families of solutions come in. Mathloves answer is detailed and potentially useful in regards to formatting the answer.

https://math.stackexchange.com/questions/3001095/integer-solutions-of-a-variable-coefficient-polynomial

Thank you for any help / advice. Ben

Given 2 inputs
m = integer
c = integer

where c ≤ m/2

and variables, a_i, b_i, a, b (all non-negative integers)
Here, m is the digit length of a number where a_i and b_i represent digits in a number writen in base b. a is simply b-1 and is used as another digit.

I am interested in generating specific permutations of a_1 ... a_c  a ... a  b_c ... b_1 where the total number of digits is m and the number of a's in the centre is m-2c.

The permutations I am interested in satisfy the condition. a_i is to the left of b_j for i ≤ j


examples for m=7:
c=1
a1,a,a,a,a,a,b1

c=2
a1,a2,a,a,a,b2,b1
a1,a2,a,a,a,b1,b2
a1,b1,a,a,a,a2,b2
a2,a1,a,a,a,b1,b2
a2,a1,a,a,a,b2,b1

c=3
a1,a2,a3,a,b3,b2,b1
...

c=4
n/a   2c > m

I would like a function L(m,c) that creates all valid permuatations of a_1 .. a_c b_1 .. b_c  (2n terms)


I met up with a lecturer at my university and we came up with this code. Unfortunately it is for mathematica as he doesn't use Maple. I feel there may be a more efficient way than to generate all permutations and then delete some.

thank you in advance for any help.

Ben

I am in the process fo creating 2 sets of multivariable expressions and I need to create pairwise equations from these sets.

L={L1(a_1 .. a_n), L2(a1, .., an) .... }

R={R1(a_1 .. a_n), R2(a1, .., an) .... }

I need to create the array:

M=Array(1..(2c)!, 1..2^c) where the elements are:

L[1] = R[1],   L[1] = R[2],   L[1] = R[3] ...
L[2] = R[1],   L[2] = R[2],   L[2] = R[3] ...
L[3] = R[1], ...


I then need to pull out particular elements (equtions M[1,1], M[7,2], etc) from the array and plug in a lot of values for the variables of that equation

for a[1] from 1 to 10 do
   for a[2] from 1 to a[1] do
       for a[3] from 1 to a[2] do
if Equation holds (store the inputs somewhere)
end do
end do
end do


I last used Maple about 5 years and have since forgotten a lot, so any and all help is greatly appreciated.
Thanks, Ben

I have a function that is a double summation and I cannot get maple to compute it. (I'm new to maple)

The function is as follows:

F(x) := sum from a=1 to x of [sum from b=0 to a-1 of [cos(2 pi x b / a)]]

for integer x, F should produce integers, but I am only getting 0's

naturally F is undefined for non integers. but maple is producing non 0 numbers for non integer entries. So, i have no idea what it is calculating.

Thanks in advance for any help. BC