[seqfan] pebbles sequence

[Tanya Khovanova]

Hello Sequence Fans,

I suggest to add pebble sequences based on the famous pebble problem:

You have 45 pebbles arranged in several piles. Each turn you take one pebble from each pile and put them into a new pile. What is an asymptotic behavior for this process? 

For example, if you start with one pile of 6, next step will be two piles: {1,5}, then {2,4} and so on. Eventually the sequence of partitions will start repeating itself.

The sequence I calculated is the following: given n - the number of pebbles we take a partition and apply the operation above until we see a partition that we saw before. The sequence is the number of steps we need until the sequence repeats itself for the worst partition.

For example, for n = 6, the worst case is {2,2,1,1}, and the steps are: {2, 2, 1, 1}, {1, 1, 4}, {3, 3}, {2, 2, 2}, {1, 1, 1, 3}, {2, 4}, {1, 
  2, 3}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3}, {1, 2, 3}. Hence a(6) = 7.

Here is the coding in Mathematica and the sequence:

In[33]:= << Combinatorica`
step[list_] := Sort[Select[Prepend[list - 1, Length[list]], # > 0 &]]
cycleStart[list_] := (res = 1; sofar = {list}; current = list; 
  nextStep = Nest[step, current, 1]; 
  While[! MemberQ[sofar, nextStep], res++; current = nextStep; 
   nextStep = Nest[step, current, 1]; sofar = Append[sofar, current]];
   res)
Table[Max[Map[cycleStart, Partitions[n]]], {n, 30}]

Out[36]= {1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 13, 13, 14, 19, 21, 21, 18, 19, 22, 29, 31, 31, 25, 25, 26, 33, 41, 43, 43, 36}

My problem is that I am not sure which sequence to call the pebble sequence, the one I calculated or a(n)+1, when we see the first repetition, or the longest sequence of terms before the cycle starts.

I need help with naming and describing these sequences, (not to mention that more terms would be nice)

Tanya