

A201144


The pebble sequence: a(n) is the length of the longest nonrepeating sequence of pebblemoves among the partitions of n.


2



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, 32, 33, 37, 46, 55, 57, 57, 49, 41, 41, 42, 51, 61, 71, 73, 73, 64, 55, 50, 51, 56, 67, 78, 89, 91, 91, 81, 71, 61, 61, 62, 73, 85, 97, 109, 111, 111, 100, 89, 78, 72, 73, 79, 92, 105, 118, 131, 133, 133, 121, 109, 97, 85, 85, 86, 99, 113, 127, 141, 155, 157, 157, 144, 131, 118, 105, 98, 99, 106, 121
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OFFSET

1,2


COMMENTS

You have n pebbles arranged in several piles. At each turn you take one pebble from each pile and put them into a new pile.
For example, if you start with one pile of 6, at the next step there are two piles: {1,5}, then {2,4}, and so on. Eventually the sequence of partitions will repeat.
Here a(n) is the maximal number of steps before a repeat among all starting partitions.


LINKS

Table of n, a(n) for n=1..100.
Jorgen Brandt, Cycles of Partitions, Proc. AMS, Vol. 85, No. 3 (July, 1982), pp. 483486
Tanya Khovanova and others, Postings to the Sequence Fans Mailing List, circa Nov 18 2009


EXAMPLE

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.


MATHEMATICA

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}


CROSSREFS

For the length of the longest cycle, see A183110. For the length of the shortest cycle, see A054531.
Sequence in context: A198045 A068998 A152854 * A095274 A079111 A007989
Adjacent sequences: A201141 A201142 A201143 * A201145 A201146 A201147


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Nov 27 2011


STATUS

approved



