searching - Find permutation index

Firstly I wanted to find all the non-negative integer solutions for the following equation given the value of $k_{max}$

$$ k_1 +k_2 +k_3+k_4 = k_{max} $$

For this I use the following code:-

findAll[kmax_]:=Partition[
 Flatten[Permutations /@ IntegerPartitions[kmax, {4}, Range[0, 
 kmax]]], 4]

Secondly I want to do in some sense the inverse of the above problem which is to find the position of the $k_i$ state given a set of $\{k_1,k_2,k_3,k_4\}$ and $k_{max}$.

For this part I use the code below.

findIndex[kmax_, state_] := Position[Partition[
Flatten[Permutations /@ IntegerPartitions[kmax, {4}, Range[0, kmax]]], 

4], val_ /; val == state]

findIndex[4, {0, 1, 1, 2}]
(*{{34}}*)

I really don't like this method for the second part.

Can this be achieved without going over all the permutations?

Answer

I think you'll find this considerably more efficient than the accepted answer:

f= With[{s = Accumulate@Reverse@# + 1, r = Range[Length@# - 1]},

         Tr[(Pochhammer[Rest@s, r] - Pochhammer[Most@s, r])/r!] + 1] &;

Example:

test={77620, 58215, 38810, 19405};

f@test // RepeatedTiming

Block[{$RecursionLimit = 100000}, index[Tr@test, Most@test] // RepeatedTiming]

{0.000017, 954814548275041}

{0.7, 954814548275041}

About 4 orders of magnitude faster.

Additionally, no need to mess with $RecursionLimit for larger cases (which even with it, can crash kernel using index, at least this happens on my machine).

Comments

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