Expand Kirkman's Schoolgirl Problem

Question

For those of you who are unfamiliar, Kirkman's Schoolgirl Problem goes as follows:

Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk twice abreast.

We could look at this like a nested 3 by 5 list (or matrix):

[[a,b,c]
 [d,e,f]
 [g,h,i]
 [j,k,l]
 [m,n,o]]

Essentially, the goal of the original problem is to figure out 7 different ways to arrange the above matrix so that two letters never share a row more than once. From MathWorld (linked above), we find this solution:

[[a,b,c]   [[a,d,h]   [[a,e,m]   [[a,f,i]   [[a,g,l]   [[a,j,n]   [[a,k,o]
 [d,e,f]    [b,e,k]    [b,h,n]    [b,l,o]    [b,d,j]    [b,i,m]    [b,f,g]
 [g,h,i]    [c,i,o]    [c,g,k]    [c,h,j]    [c,f,m]    [c,e,l]    [c,d,n]
 [j,k,l]    [f,l,n]    [d,i,l]    [d,k,m]    [e,h,o]    [d,o,g]    [e,i,j]
 [m,n,o]]   [g,j,m]]   [f,j,o]]   [e,g,n]]   [i,k,n]]   [f,h,k]]   [h,l,m]]

Now, what if there were a different number of schoolgirls? Could there be an eighth day?^† This is our challenge.

_{^†In this case no^††, but not necessarily for other array dimensions
^††We can easily show this, since a appears in a row with every other letter.}

---

The Challenge:

Given an input of dimensions (rows, than columns) of an array of schoolgirls (i.e. 3 x 5, 4 x 4, or [7,6], [10,10], etc.), output the largest possible set of 'days' that fit the requirements specified above.

Input:
The dimensions for the schoolgirl array (any reasonable input form you wish).

Output:
The largest possible series of arrays fitting the above requirements (any reasonable form).

Test Cases:

Input:  [1,1]
Output: [[a]]

Input:  [1,2]
Output: [[a,b]]

Input:* [2,1]
Output: [[a]
         [b]]

Input:  [2,2]
Output: [[a,b]  [[a,c]  [[a,d]
         [c,d]]  [b,d]]  [b,c]]

Input:  [3,3]
Output: [[a,b,c]  [[a,d,g]  [[a,e,i]  [[a,f,h]
         [d,e,f]   [b,e,h]   [b,f,g]   [b,d,i]
         [g,h,i]]  [c,f,i]]  [c,d,h]]  [c,e,g]]

Input:  [5,3]
Output: [[a,b,c]   [[a,d,h]   [[a,e,m]   [[a,f,i]   [[a,g,l]   [[a,j,n]   [[a,k,o]
         [d,e,f]    [b,e,k]    [b,h,n]    [b,l,o]    [b,d,j]    [b,i,m]    [b,f,g]
         [g,h,i]    [c,i,o]    [c,g,k]    [c,h,j]    [c,f,m]    [c,e,l]    [c,d,n]
         [j,k,l]    [f,l,n]    [d,i,l]    [d,k,m]    [e,h,o]    [d,o,g]    [e,i,j]
         [m,n,o]]   [g,j,m]]   [f,j,o]]   [e,g,n]]   [i,k,n]]   [f,h,k]]   [h,l,m]]

There may be more than one correct answer. 

_{*Thanks to @Frozenfrank for correcting test case 3: if there is only one column, there can only be one day, since row order does not matter.}

This is code-golf competition - shortest answer wins.

Answer 1

Mathematica, 935 bytes

Inp={5,4};L=Length;T=Table;ST[t_,k_,n_]:=Binomial[n-1,t-1]/Binomial[k-1,t-1];H=ToExpression@Alphabet[];Lo=Inp[[1]]*Inp[[2]];H=H[[;;Lo]];Final={};ST[2,3,12]=4;ST[2,4,20]=5;If[Inp[[2]]==1,Column[Partition[H,{1}]],CA=Lo*Floor@ST[2,Inp[[2]],Lo];While[L@Flatten@Final!=CA,Final={};uu=0;S=Normal[Association[T[ToRules[H[[Z]]==Prime[Z]],{Z,L@H}]]];PA=Union[Sort/@Permutations[H,{Inp[[2]]}]];PT=Partition[H,Inp[[2]]];While[L@PA!=0,AppendTo[Final,PT];Test=Flatten@T[Times@@@Subsets[PT[[X]],{2}]/.S,{X, L@PT}];POK=T[Times@@@Subsets[PA[[Y]],{2}]/.S,{Y,L@PA}];Fin=Select[POK,L@Intersection[Test,#]==0&];Facfin=T[FactorInteger[Fin[[V]]],{V,L@Fin}];end=T[Union@Flatten@T[First/@#[[W]],{W,L@#}]&[Facfin[[F]]],{F,L@Facfin}]/.Map[Reverse,S];PA=end;PT=DeleteDuplicates[RandomSample@end,Intersection@##=!={}&];If[L@Flatten@PT<L@H,While[uu<1000,PT=DeleteDuplicates[RandomSample@end,Intersection@##=!={}&];If[L@Flatten@PT==L@H,Break[],uu++]]]]];Grid@Final]

this is for 26 ladies max

EDIT
I made some changes and I think it works! The code right now is set to solve [5,4] (which is the "social golfers problem") and gets the result in a few seconds. However [5,3] problem is tougher and you will have to wait 10-20 minutes but you will get a right combination for all days. For easier cases it is very quick.

anyway you can try it and see the results
Try it online here
copy and paste using ctrl-v
press shift+enter to run the code
you can change the input at the begining of the code -> Inp={5,4}
run the code multiple times to get different permutations

Answer 2

Jelly, 19 bytes

׌!s€ɗœcⱮ×ẎẎṢ€QƑƊƇṪ

Try it online!

The Footer simply formats the input and output and demonstrates for various inputs. Assumes that it's ok to return any valid configuration, rather than the one stated in the question.

This is a brute force approach and isn't efficient by any meaning of the word. If we call the dimensions \$n\$ and \$m\$ (\$n\$ being height), this generates

$$f(n, m) = \sum^{nm}_{i=1} \binom {(nm)!} {i}$$

lists to check. For example, for \$n = 3, m = 5\$, we generate $$42755752626996476442177192901690329126386272975917980316235035492211700060653916251199050827754168292191219682093923414635027126605138973495697914193291687743693535776000$$

lists to check. Unsurprisingly, this times out on TIO for basically all inputs. Those for which it doesn't time out are shown in the TIO link.

How it works

׌!s€ɗœcⱮ×ẎẎṢ€QƑƊƇṪ - Main link. Takes n on the left and m on the right
     ɗ              - Group the previous 3 links into a dyad f(n, m):
×                   -   Yield n×m
 Œ!                 -   Promote n×m into a range [1, 2, ..., n×m] and yield
                         all permutations of that range
    €               -   Over each permutation:
   s                -     Split it into an n×m matrix
         ×          - Yield n×m
        Ɱ           - For each integer 1 ≤ i ≤ n×m:
      œc            -   Yield all combinations (without replacement) of the
                         permutations of length i
          Ẏ         - Tighten these into a single list of lists of matrices
                ƊƇ  - Keep those for which the following is true:
              QƑ    -   The matrix has no duplicates when:
           Ẏ        -     Tightened into a list of lists
            Ṣ€      -     And each list is sorted
                  Ṫ - Get the last (i.e. the longest) such list