Suitable subtle subtables

We can delete, or just simply ignore, arbitrary rows and columns of a given table.  The result is a subtable.  Any subtable of a proportion table is also a proportion table.  Here, we delete the third row, and the first and third columns, of a table to get a subtable :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27)] \qquad->\qquad [(4,  9),(8, 18)] `.

In general, making a subtable is a lossy operation---there is less table when we are done.

We can split a table into a complete, nonoverlapping collection of subtables.  For instance :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(2, \qquad4),(4, \qquad8)]\qquad [(5, 9),(10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :} `.

The preceding patchwork of subtables was made by carving up the original table in an obvious way.  A nonoverlapping collection does not have to fit together like patchwork, however :

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(\quad2, 9),(10, 45)]\qquad [(4, 5),(20, 25)]), ([(\quad4, 18),(\quad6, 27)]\qquad [(8, 10),(12, 15)]) :} `.

The subtables do not have to be square, or the same size, or deployed in a symmetrical pattern.  The most useful decompositions are those that are tartanlike, meaning that any split between cells belonging to distinct subtables goes all the way across the original table.

The two preceding decompositions are tartanlike.  This one is not:

`\qquad\qquad [(2, 4, 5, 9),(4, 8, 10, 18),(6, 12, 15, 27), (10, 20, 25, 45)] \qquad->\qquad{: ([(2, \qquad4,\qquad5, 9),(4, \qquad8,\qquad10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :} `.

We have arranged the subtables according to the position of its reference element---by convention its top left element---in the original table.  Using such a convention is no mere convenience, as we shall soon see.

Tartanlike decompositions can be aggregated, by adding together the elements in each subtable :

`\qquad\qquad\qquad{: ([(2, \qquad4),(4, \qquad8)]\qquad [(5, 9),(10, 18)]), ([(6, 12),(10, 20)]\qquad [(15, 27),(25, 45)]) :}\quad->\quad{: ([(),(\qquad18\qquad),()]\quad [(),(\qquad42\qquad),()]), ([(),(\qquad48\qquad),()]\quad [(),(\quad112\qquad),()]) :}`,

where the result can also be written

`\qquad\qquad\qquad[(18,42),(48,112)]`.

Similarly,

`\qquad\qquad\qquad{: ([(\quad2, 9),(10, 45)]\qquad [(4, 5),(20, 25)]), ([(\quad4, 18),(\quad6, 27)]\qquad [(8, 10),(12, 15)]) :}\quad->\quad[(66,54),(55,45)] `,

where the convention we used before helps us keep the aggregated subtables in proportional relation.

A table filled entirely with ones is a proportion table.  By aggregating such a table of suitable size, we can construct any proportion table whatever that has only positive integer elements.  For instance :

`\qquad\qquad\qquad [(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1),(1,1,1,1,1,1)]\quad->\quad[(1,2,3),(2,4,6),(3,6,9)],

where an appropriate tartanlike decomposition that leads from the first of these tables to the second should be obvious.

These three examples illustrate a general truth: aggregating a proportion table always gives another proportion table.