[seqfan] Re: Partitions and Dice

[franktaw at netscape.net]

OK, this looks correct.  The original comment did lack the "off by 1" 
part.

This leads me to wonder: is t he number of partitions of n into k 
parts, each <= k, always equal to the number of partitions of n-1 into 
k-1 parts, each <= k+1?  This is the case k=5; I checked k=2 (trivial) 
and k=3.  So it seems very likely to me that these are always the same. 
  I don't see any simple correspondence, though -- if the partition into 
k parts includes a 1, you can simply remove it; but there is no obvious 
transformation for the partitions that do not include a 1.  Can anyone 
supply a proof, here?

Franklin T. Adams-Watters

-----Original Message-----
From: Toby <toby at gottfriedville.net>

----- Original Message ----- 
Subject: Partitions and Dice 
 

Second, the comment from Toby Gottfried in A102422 is fine, but the one 
in A102420 is not correct.  This is 5 "5-sided" dice, not 4 6-sided 
dice. 

Franklin T. Adams-Watters 
---------------------------------------------------------------- 
 
It's both. 

Same numbers work for two different criteria. 

A102420      Number of partitions of n with exactly k = 5 parts and 
each part p <= 5 

New:  exactly 4 parts and 1 <= p <= 6  ... regular dice  (and different 
offset by 1 in
sequence) 

Example: total of 7 
exactly 4 "hexa dice" 

4 1 1 1 0D
3 2 1 1 
2 2 2 1 

---- 3 partitions 
 
total of 8 
5 penta dice 

4 1 1 1 1 
3 2 1 1 1 
2 2 2 1 1 

---- 3 partitions 

>> Same number of partitions, 
different total by 1 
different number of dice 
different max number on dice.