Note that 286 is Binomial[13,3]
. This is not a coincidence. The latter is the number of ways one can place 3 sticks around or between 10 stones (a classic combinatorial problem). One gets from each such placement a partition by taking the first element with multiplicity equal to number of stones before (to the left of) the first stick (this number could be zero), the second element is repeated the number of stones between stick 1 and stick 2, with the third element repeated the number of stones between stick 2 and stick 3 and the fourth element repeated the number of stones after stick 3, Apologies for that belabored explanation
So here is some code. First get the stick placements.
Length[tt = Flatten[Table[{i, j, k}, {k, 3, 13}, {j, 2, k - 1}, {i, 1, j - 1}], 2]]
Short[tt]
(* Out[30]= 286
{{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 5}, << 277 >>, {8,
12, 13}, {9, 12, 13}, {10, 12, 13}, {11, 12, 13}} *)
Now code to transform a placement into a partition of {0,1,2,3}
into an ordered set of 10 elements.
partition[oll_List] := Module[{ll = Flatten[{0, oll, 14}], diffs},
diffs = Differences[ll] - 1;
Join @@ Table[ConstantArray[j - 1, diffs[[j]]], {j, Length[diffs]}]]
tuples = Map[partition, tt];
Short[tuples, 12]
(* {{3,3,3,3,3,3,3,3,3,3},{2,3,3,3,3,3,3,3,3,3},{1,3,3,3,3,3,3,3,3,3},{0,3,3,3,3,3,3,3,3,3},
{2,2,3,3,3,3,3,3,3,3},{1,2,3,3,3,3,3,3,3,3},{0,2,3,3,3,3,3,3,3,3},{1,1,3,3,3,3,3,3,3,3},
{0,1,3,3,3,3,3,3,3,3},{0,0,3,3,3,3,3,3,3,3},{2,2,2,3,3,3,3,3,3,3},{1,2,2,3,3,3,3,3,3,3},<<263>>,
{1,1,1,1,1,1,1,1,1,1},{0,1,1,1,1,1,1,1,1,1},{0,0,1,1,1,1,1,1,1,1},{0,0,0,1,1,1,1,1,1,1},
{0,0,0,0,1,1,1,1,1,1},{0,0,0,0,0,1,1,1,1,1},{0,0,0,0,0,0,1,1,1,1},{0,0,0,0,0,0,0,1,1,1},
{0,0,0,0,0,0,0,0,1,1},{0,0,0,0,0,0,0,0,0,1},{0,0,0,0,0,0,0,0,0,0}} *)
Add spice, adjust to taste.