[seqfan] Re: pebbles sequence

[franktaw at netscape.net]

Each of these is actually two variants.  One is exactly as Max 
describes it, the other inserts the new pile into its place so that the 
pile sizes remain decreasing (thus being an operation on partitions 
instead of on compositions).

Franklin T. Adams-Watters

-----Original Message-----
From: Max Alekseyev <maxale at gmail.com>

...

There is an interesting variant of this problem: Suppose we have k
(nonempty) piles with pebbles arranged in a row. Each turn one take
all, say, m pebbles from the leftmost pile and spread them equally
among the other k-1 piles (i.e., each pile gets [m/(k-1)] new
pebbles), putting the remaining m mod (k-1) pebbles (if any) into a
new rightmost pile. The process stop when we either encounter a cycle
or a single pile.

E.g.: (5,5,5,5) -> (6,6,6,2) -> (8,8,4) -> (12,8) -> (20)   (a single 
pile)

E.g.: (5,3,1) -> (5,3,1) -> ...  (a cycle of length 1)

Other variant: Each turn distribute m pebbles from the leftmost pile
over the other piles from left to right, putting one pebble into each
pile (i.e., each pile gets at most one new pebble), and the remainder
m-(k-1) (if it's nonnegative) into a new rightmost pile.

E.g.: (5,5,5,5) -> (6,6,6,2) -> (7,7,3,3) -> (8,4,4,4) -> (5,5,5,5)
(a cycle of length 4)

Somewhat similar game is described in
http://www.research.att.com/~njas/sequences/A161135

Regards,
Max


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