Nice mapping between odd partitions and distinct
partitions.
Reminds me of the slick mapping between distinct odd
partitions and self-dual
partitions. Bend the odd numbers into gnomons and nest them:
Distinct odd partition (7, 5, 1)
A
A
AB
AB
AB
AB
ABC
maps to self-dual partition (4, 4, 3, 2)
AB
ABC
ABBB
AAAA
----- Original Message -----
Sent: Thursday, May 15, 2003 3:45 PM
Subject: Re: [math-fun] distinct and square-free
factorizations
[omitted stuff]
The combinatorial proof shows a direct 1-1
correspondence between
partitions of the two types. To convert a
partition of the "odd" type to
one of the "distinct" type, find a pair of
identical piles and merge
them. Repeat until there are no identical
piles; by construction the
resulting partition is of the "distinct"
type. The inverse operation is to
find an even pile and break it into
two equal ones. This operation is
repeated until there are no even
piles, so the result is a partition of
"odd" type.
[more omitted
stuff]
-A