This does it! You are way ahead of me, Udo. Up to me now to do some further experimenting.

Thank you Udo, thank you Todd. With your observations in mind, I started to realise that:

1. there is a problem with all values of n but less so with smaller values
2. there is indeed something with the iterator i. I started noticing that there is a discrepancy between the evaluation speed of the If[] and the subsequent BarChart[]. Furthermore, using (the faster?) ArrayChart gives less of a problem. I tried several things such as SynchronousUpdating etc but the only thing that worked is to set the animation refresh rate with AnimationRate->1 or <1. Higher than 1 or Automatic does not give (always) a correct sort.

Here is a gif of a working example with the added AnimationRate->1:

It works but I do not completely understand why. It must have something to do with Evaluation speed or evaluation order...?

Start the manipulation with an ordered set to see that only the first two entries become interchanged forever: $i$ does not escalate beyond 2 ...

Manipulate[Module[{pivot, sample},
      pivot = n/2; sample = lst[[i]];
      If[sample < pivot,
       PrependTo[lst, sample]; lst = Delete[lst, i + 1]]; 
      BarChart[lst, GridLines -> {None, {pivot}}, BarSpacing -> 1]],
     Row[{Control[{{i, 1}, 1, n, 1, Trigger, DisplayAllSteps -> True, 
         AnimationRunning -> False, AnimationRepetitions -> 1}], 
       Dynamic@i}],
     {{n, 30}, None},

 Initialization :> (lst =(* RandomSample[Range[n],n]) *) Range[n]) ]

The iteration took place if I resized the notebook!

Manipulate[With[{pivot = n/2, cut = n/2, sample = lst[[i]]},
      If[sample < pivot,
       lst = Join[RotateRight[Take[lst, i]], Take[lst, i - n]]
      ];
      BarChart[lst, GridLines -> {None, {pivot}}, BarSpacing -> 1]],
     Row[{Control[{{i, 1}, 1, n, 1, Trigger, DisplayAllSteps -> True, 
         AnimationRunning -> False, AnimationRepetitions -> 1}], 
       Dynamic@i}],
     {{n, 50}, None},
     Initialization :> (lst =(* RandomSample[Range[n],n] *) Range[n])]

I suspect there is a problem with the iterator $i$.

Thanks for your observation, Udo. I am aware of the n=3 being hard coded but (maybe I explained it wrongly?) my problem is that when I (again hard coded) change to {{n,50},None}, I get no longer a separation between the bars larger than and smaller than the pivot. As can be seen from this gif with n=30 first where the sort is ok and the bars are neatly separated:

and now with n=50 where I have a problem as you can see:

I realise that a Manipulate/Trigger combination may be the wrong way to do this but there must be something I am missing here and I want to understand. Maybe you can help with your expertise. By the way, love yr website zahlenteppich!

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