Cfrtjryk

So I would like to solve the following set of equation for $m_i$ given a set of ${M_m,N_m}$.

$$
m_1 +m_2 +m_3 +m_4 =M_m \
|m_1| +|m_2| +|m_3| +|m_4| =N_m
$$

All variables are integers.
Also $N_m ge M_m$ and their maximum value can reach up-to 30.
I only need the total number of possible solution not the solutions themselves. So my first trivial attempt was to just use Solve

dimNM1[Nm_, Mm_] :=

Length[(Solve[m1 + m2 + m3 + m4 == Mm && 

 Abs[m1] + Abs[m2] + Abs[m3] + Abs[m4] == Nm, {m1, m2, m3, m4}, Integers])]

My second slightly non-trivial attempt is the following:-

dimNM2[Nm_, Mm_] :=

Which[Nm === Mm, 

  Length[Partition[

Flatten[Permutations /@ IntegerPartitions[Nm, {4}, Range[0, Nm]]], 

4]], True,

 Module[{res},

res = Partition[

 Flatten[Permutations /@ IntegerPartitions[Mm, {4}, Range[-Nm, Nm]]],

  4];

  Length[

Select[res, (Abs[#[[1]]] + Abs[#[[2]]] + Abs[#[[3]]] + 

     Abs[#[[4]]]) == Nm &]]]]

The second method is much faster than the first specially for $N_m=M_m$.
But I would like to increase the speed further for $N_mge M_m$ case if possible.

dimNM1[2, 2] // AbsoluteTiming

(*{0.177768, 10}*)



dimNM2[2, 2] // AbsoluteTiming

(*{0.0000899056, 10}*)

So is there any other way to solve these equation faster?

It is more efficient to first pick the integer partions whose absolute values sum up to n before generating the permutations.

dimNM3[n_, m_] := Total[

   Map[

    Length@*Permutations,

    Pick[#, Abs[#].ConstantArray[1, 4], n] &[

     IntegerPartitions[m, {4}, Range[-n, n]

      ]

     ]

    ]

   ];



m = 20;

n = 40;

dimNM1[n, m] // AbsoluteTiming

dimNM2[n, m] // AbsoluteTiming

dimNM3[n, m] // AbsoluteTiming

{0.116977, 3802}

{0.995365, 3802}

{0.005579, 3802}