[seqfan] Re: need some help with partitions (cores and quotients)

Wouter Meeussen wouter.meeussen at telenet.be
Thu Jun 9 01:15:16 CEST 2016


thanks Keith,

gimme a while to fully absorb this. (like a month or so, LOL)
If I get stuck, I'll come back to you privately.
Is that ok? (no need to bother innocent folks with this)


Wouter.



-----Original Message----- 
From: William Keith 
Sent: Wednesday, June 08, 2016 10:06 PM 
To: Sequence Fanatics Discussion list 
Subject: [seqfan] Re: need some help with partitions (cores and quotients) 

They way I find best to understand the construction of cores and quotients
is to use the abacus visualization.  This takes the profile of the Ferrers
diagram of a partition, usually from the 4th quadrant (I will use English
style) and replaces it with beads (I will use *) for vertical steps (tops
of parts) and spacers (I will use o ) for horizontal steps (increasing size
of parts) on horizontal runners of length p.  The profile of partition
begins with an infinite sequence of vertical steps marked by beads,
replaces any horizontal steps with spacers, and ends with a final bead
followed by an infinite sequence of spacers.

Here you decompose {5, 3, 2, 1, 1, 1} into core {2, 1, 1} and quotients {},
{1}, {1, 1}, which means you are using p=3.  The 3-abacus of {5,3,2,1,1,1}
is

***  <- infinite depth of these above
o**
*o*
o*o
o*o
ooo <- infinite tail of these below

A t-core partition must have an abacus consisting of t uninterrupted
columns of beads followed by uninterrupted columns of spacers, because a
spacer followed by a bead on the next runner yields a t-hook.  Removing
t-hooks from a partition corresponds to collapsing the beads in a colum up
one row, removing a spacer; repeatedly doing so transforms the abacus in to
one with no internal spacers, yielding a core partition.  In this case,
pushing all beads up yields

*** <- infinite sequence
***
o**
o*o
ooo <- infinite sequence

This is, indeed, the profile of {2,1,1} (vertical quadrant edge, horizontal
step, vertical step, vertical, horizontal, vertical, horizontal quadrant
edge).  Conventionally we put the first spacer in the first column, but we
do not have to: the profile of the positive parts begins wherever the first
spacer is.

The quotient components are the three partitions which we obtain if we read
each column as if it were the profile of partition, again with an infinite
prior sequence of beads, starting on positive parts at the first spacer,
and ending with an infinite sequence of spacers.  Going back to the
original abacus, we see that these are, from left to right, (o*), (0**),
and (), which are your three quotient components.

To reverse the process, begin with the profile of your core partition.  The
first spacer should go in the first column.  Note that in order to remove
ambiguity, the core and quotients must be reported so that when this is
done, the first quotient component comes from the column with this spacer.

Now push beads downward so that the abaci in each column below the solid
initial sequence of beads match the corresponding quotient component.  The
result will be the profile of the original partition.

Cordially,
William J. Keith

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