Sort nested list by a fixed order

Question

I have a nested list like this one:

list={{{a,1},{b,3},{c,5}},{{b,1},{c,3},{a,5}},{{c,1},{b,3},{a,5}},{{a,1},{c,3},{b,5}}}

Now I need to sort the list such that the order within a row is always {{a,..},{b,..},{c,..}}. Put differently: the order of the first elements within the sublists should always be a, b and then c.

Answer 1

ClearAll[sortLike]
sortLike[refcolumn_, orderlike_] := 
 Map[#[[Ordering[#[[All, refcolumn]]][[Ordering @ Ordering @ orderlike]]]] &]

Examples:

list = {{{a, 1}, {b, 3}, {c, 5}}, {{b, 1}, {c, 3}, {a, 5}}, 
   {{c, 1}, {b, 3}, {a, 5}}, {{a, 1}, {c, 3}, {b, 5}}};


sortLike[1, {a, b, c}]@list // Column

sortLike[1, {b, a, c}]@list // Column

sortLike[2, {1, 2, 10}]@list // Column

sortLike[2, {5, 1, 3}]@list // Column

The argument orderlike can be given alternative ways to get the same result:

Multicolumn[Labeled[Column[sortLike[1, #]@list], 
    Row[{"orderlike: ", Style[#, ShowStringCharacters -> True]}], Top] & /@ 
 {{c, a, b}, {"c", "a", "b"}, {"FOO", "BAR", "BUZZ"}, {3, 1, 2}, 
  {100, 0, 25}, foo[10, 1, 9]}, 3,
 Dividers -> All, Alignment -> Center]

Answer 2

Something like the following:

Map[Sort[#] &, list]
(*{{{a, 1}, {b, 3}, {c, 5}}, {{a, 5}, {b, 1}, {c, 3}}, {{a, 5}, {b, 3}, {c, 1}}, {{a, 1}, {b, 5}, {c, 3}}}*)

A first approximation:

MyOrderList[list_List, order_?(Positive[#] && Element[#, Integers] &)] := 
Block[{slist, perm, mylist},
slist := Map[Sort, list];
perm = Mean[Map[Composition[Length, Permutations[#] &], slist]];
mylist := 
Table[Table[Extract[Select[Tuples[slist[[i]], Length[slist[[i]]]], 
ContainsAll[#, slist[[i]]] &], j], {i, 1, Length[slist]}], {j, 1, perm}][[order]];
Return[If[order <= perm, mylist, HoldForm[MyOrderList]]];
];

Tests:

MyOrderList[list, 1]
(*{{{a, 1}, {b, 3}, {c, 5}}, {{a, 5}, {b, 1}, {c, 3}}, {{a, 5}, 
  {b, 3}, {c, 1}}, {{a, 1}, {b, 5}, {c, 3}}}*)
MyOrderList[list, 2]
(*{{{a, 1}, {c, 5}, {b, 3}}, {{a, 5}, {c, 3}, {b, 1}}, {{a, 5}, 
  {c, 1}, {b, 3}}, {{a, 1}, {c, 3}, {b, 5}}}*)
MyOrderList[list, 3]
(*{{{b, 3}, {a, 1}, {c, 5}}, {{b, 1}, {a, 5}, {c, 3}}, {{b, 3}, 
  {a, 5}, {c, 1}}, {{b, 5}, {a, 1}, {c, 3}}}*)
MyOrderList[list,4]
(*{{{b, 3}, {c, 5}, {a, 1}}, {{b, 1}, {c, 3}, {a, 5}}, {{b, 3}, 
  {c, 1}, {a, 5}}, {{b, 5}, {c, 3}, {a, 1}}}*)
MyOrderList[list,5]
(*{{{c, 5}, {a, 1}, {b, 3}}, {{c, 3}, {a, 5}, {b, 1}}, {{c, 1}, 
  {a, 5}, {b, 3}}, {{c, 3}, {a, 1}, {b, 5}}}*)
MyOrderList[list,6]
(*{{{c, 5}, {b, 3}, {a, 1}}, {{c, 3}, {b, 1}, {a, 5}}, {{c, 1}, 
  {b, 3}, {a, 5}}, {{c, 3}, {b, 5}, {a, 1}}}*)

Another approach using the idea of @cvgmt:

  MyOrderList[list_List, order_List] := 
  Block[{slist, sorder, ordering, mylist},
  slist := Map[Sort, list];
  ordering := 
  Extract[Permute[Ordering[Sort[order]], Ordering[#]] & /@ 
  Permutations[order], {1}];(*@cvgmt*)
  mylist := Table[slist[[i]][[ordering]], {i, 1, Length[slist]}];
  Return[mylist];
   ];

Test:

MyOrderList[list, {b, c, a}]
(*{{{b, 3}, {c, 5}, {a, 1}}, {{b, 1}, {c, 3}, {a, 5}}, {{b, 3}, 
  {c, 1}, {a, 5}}, {{b, 5}, {c, 3}, {a, 1}}}*)

Another approach using the idea of @kglr:

 MyOrderList[list_List, order_List] := 
 Block[{slist, sorder, ordering, mylist},
 slist := Map[Sort, list];
 ordering := 
 Extract[Map[Ordering@*Ordering, Permutations[order]], {1}];(*@kglr*)
 mylist := Table[slist[[i]][[ordering]], {i, 1, Length[slist]}];
 Return[mylist];
   ];

Test:

MyOrderList[list, {c, a, b}]
(*{{{c, 5}, {a, 1}, {b, 3}}, {{c, 3}, {a, 5}, {b, 1}}, {{c, 1}, 
  {a, 5}, {b, 3}}, {{c, 3}, {a, 1}, {b, 5}}}*)