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list manipulation - Equivalent Nested Loop Structure


Consider the following examples:


Flatten @ Table[ f[a, b, c], {a, 1, 3}, {b, 1, 3}, {c, 1, 3}] ==
f @@@ Tuples[ Range[3], 3]

Flatten @ Table[ If[ a != b != c, f[a, b, c], Unevaluated[]], 
                 {a, 1, 3}, {b,  1, 3}, {c, 1, 3}] == 
f @@@ Permutations[ Range[3], {3}]


Flatten @ Table[ f[a, b, c], {a, 1, 5}, {b, a + 1, 5}, {c, b + 1, 5}] == 
f @@@ Subsets[ Range[5], {3}]


True
True
True

Is there a similar built-in function performing this:



Flatten @ Table[ f[a, b, c], {a, 1, 5}, {b, a, 5}, {c, b, 5}]

in a simpler way?



Answer



Here a two more ways, neither of them built-in.
f @@ (# - {0,1,2})& /@ Subsets[Range@7,{3}]
f @@@ Transpose[Transpose@Subsets[Range@7,{3}] - {0,1,2}]


EDIT - The question is related to a long and somewhat tangled MathGroup thread generating submultisets with repeated elements. "sms" below stands for for "submultisets".


sms[n_, k_, f_:List] := With[{i = Table[Unique[],{k}]}, Flatten[Table[f@@i,
   Evaluate[Sequence@@Transpose@{i,Prepend[Most@i,1],Table[n,{k}]}]],k-1]]


sms[5, 3]
sms[5, 3, f]


{{1,1,1}, {1,1,2}, {1,1,3}, {1,1,4}, {1,1,5}, {1,2,2}, {1,2,3}, {1,2,4}, {1,2,5}, {1,3,3}, {1,3,4}, {1,3,5}, {1,4,4}, {1,4,5}, {1,5,5}, {2,2,2}, {2,2,3}, {2,2,4}, {2,2,5}, {2,3,3}, {2,3,4}, {2,3,5}, {2,4,4}, {2,4,5}, {2,5,5}, {3,3,3}, {3,3,4}, {3,3,5}, {3,4,4}, {3,4,5}, {3,5,5}, {4,4,4}, {4,4,5}, {4,5,5}, {5,5,5}}


{f[1,1,1], f[1,1,2], f[1,1,3], f[1,1,4], f[1,1,5], f[1,2,2], f[1,2,3], f[1,2,4], f[1,2,5], f[1,3,3], f[1,3,4], f[1,3,5], f[1,4,4], f[1,4,5], f[1,5,5], f[2,2,2], f[2,2,3], f[2,2,4], f[2,2,5], f[2,3,3], f[2,3,4], f[2,3,5], f[2,4,4], f[2,4,5], f[2,5,5], f[3,3,3], f[3,3,4], f[3,3,5], f[3,4,4], f[3,4,5], f[3,5,5], f[4,4,4], f[4,4,5], f[4,5,5], f[5,5,5]}



sms[data_List, k_, f_:List] := With[{i = Table[Unique[],{k}]}, Flatten[Table[f@@data[[i]],
   Evaluate[Sequence@@Transpose@{i,Prepend[Most@i,1],Table[Length@data,{k}]}]],k-1]]


data = {a,b,c,d,e};
sms[data, 3]
sms[data, 3, f]


{{a,a,a}, {a,a,b}, {a,a,c}, {a,a,d}, {a,a,e}, {a,b,b}, {a,b,c}, {a,b,d}, {a,b,e}, {a,c,c}, {a,c,d}, {a,c,e}, {a,d,d}, {a,d,e}, {a,e,e}, {b,b,b}, {b,b,c}, {b,b,d}, {b,b,e}, {b,c,c}, {b,c,d}, {b,c,e}, {b,d,d}, {b,d,e}, {b,e,e}, {c,c,c}, {c,c,d}, {c,c,e}, {c,d,d}, {c,d,e}, {c,e,e}, {d,d,d}, {d,d,e}, {d,e,e}, {e,e,e}}


{f[a,a,a], f[a,a,b], f[a,a,c], f[a,a,d], f[a,a,e], f[a,b,b], f[a,b,c], f[a,b,d], f[a,b,e], f[a,c,c], f[a,c,d], f[a,c,e], f[a,d,d], f[a,d,e], f[a,e,e], f[b,b,b], f[b,b,c], f[b,b,d], f[b,b,e], f[b,c,c], f[b,c,d], f[b,c,e], f[b,d,d], f[b,d,e], f[b,e,e], f[c,c,c], f[c,c,d], f[c,c,e], f[c,d,d], f[c,d,e], f[c,e,e], f[d,d,d], f[d,d,e], f[d,e,e], f[e,e,e]}



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