Sort sequences by their "randomness"

Question

Example matrix (rows always have equal lengths)

mat = {
   {1, 2, 3, 4, 5},
   {1, 1, 1, 1, 1},
   {1, 3, 4, 7, 9},
   {2, 2, 2, 2, 2},
   {5, 4, 3, 2, 1},
   {8, 4, 7, 9, 2},
   {3, 2, 1, 2, 3},
   {1, 3, 5, 7, 9}};

I tried to sort mat in the following manner:

{{1, 1, 1, 1, 1}, (* All elements are equal *)
 {2, 2, 2, 2, 2}, (* All elements are equal *)
 {1, 2, 3, 4, 5}, (* Sorted sequence with equal differences of 1 *)
 {5, 4, 3, 2, 1}, (* Reverse sorted sequence with equal differences of 1 *)
 {1, 3, 5, 7, 9}, (* Sorted sequence with equal differences of 2 *)
 {1, 3, 4, 7, 9}, (* Sorted sequence with unequal differences *)
 {3, 2, 1, 2, 3}, (* Unsorted sequence with few sort swaps needed *)
 {8, 4, 7, 9, 2}} (* Unsorted sequence with many sort swaps needed *)

The closest solution I found was:

SortBy[mat, Abs @* Differences]

giving

{{1, 1, 1, 1, 1},
 {2, 2, 2, 2, 2},
 {1, 2, 3, 4, 5},
 {3, 2, 1, 2, 3},
 {5, 4, 3, 2, 1},
 {1, 3, 4, 7, 9},
 {1, 3, 5, 7, 9},
 {8, 4, 7, 9, 2}}
 

Looking at {3, 2, 1, 2, 3}, which is in the wrong position, it becomes clear that we need at least one additional sort parameter for unsorted sequences: Count the number of permutations needed to Sort or ReverseSort them, but I don't know how to implement this.

Probably my description of the problem is incomplete or ambiguous, so please feel free to suggest another "entropic sort order".

Answer 1

Using Entropy:

SortBy[mat, Entropy]

{{1, 1, 1, 1, 1}, {2, 2, 2, 2, 2}, {1, 2, 3, 4, 5}, {1, 3, 4, 7,
9}, {1, 3, 5, 7, 9}, {5, 4, 3, 2, 1}, {8, 4, 7, 9, 2}, {3, 2, 1, 2,
3}}

Another attempt (and comparison) with the differences included:

{SortBy[mat, Entropy], SortBy[mat, {Entropy, Abs@*Differences}]} // 
 Map[Grid]

Answer 2

Variability of Differences of the input list seems to underly your verbal description of how desired output is obtained.

So, using Variance @* Differences (or MeanDeviation @* Differences or MedianDeviation @* Differences ...) as the first sorting criterion and Mean (Median or Variance also work for this case) as the second criterion gives the desired result:

sf = {Variance @* Differences, Mean};

Row[#, Spacer[20]] & @ Map[Labeled[MatrixForm@ToExpression@#, #, Top] &]@
 {"mat", "desired", "SortBy[sf] @ mat"}