Permutations with conditions: element a to appear before c

Questions
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1

Hello!

i need to calculate something in Mathematica;
We have the list

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

I need to find all permutations which will contain exactly 3 elements from A in which:

  1. element b appears only once (or does not appear at all)
  2. if a and c appear, a will always appear before c

So far i have:

A={a,b,c,d,e};
Select[Tuples[A, 3], OrderedQ]

But it seems i’m missing quite a lot ..
Any help will be appreciated.
Thanks!

2

There might be a clever way to do this, but let’s focus on defining functions that test your two properties

testA[list_] := Count[list, b]<2

testB[list_] := If[MemberQ[list, a] && MemberQ[list, b], 
                   First@FirstPosition[list, a] < First@FirstPosition[list, b], 
                   True]

You can use these functions with Select

3

A couple of ways come to mind. Both delete the cases that are not wanted from the output of the permutation generator.

aa = {a, b, c, d, e};
p1 = DeleteCases[
  Tuples[{aa, aa, aa}], {___, c, ___, a, ___} | {___, b, ___, b, ___}];
bb = {a, a, a, b, c, c, c, d, d, d, e, e, e};
p2 = DeleteCases[Permutations[bb, {3}], {___, c, ___, a, ___}];

p1 and p2 are the same list.

4

I believe this works too

Clear[a, b, c, d, e]; Sort /@ 
 Select[Tuples[{a, b, c, d, e}, {3}], Count[#, b] <= 1 &]