# [seqfan] Re: Partitions and Dice

franktaw at netscape.net franktaw at netscape.net
Sat Feb 21 10:37:45 CET 2009

```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?

-----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.

----------------------------------------------------------------

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.

```