partitions whose FerrersPlot just fits inside a square

[Wouter Meeussen]

hi,

a nice way to subdivide the partitions of n :
partitions of n whose FerrersPlot just fits inside a square with side m:

example: a partition of 14 : {5,3,3,1,1,1}
has FerrersPlot:

  * * * * *
  * * *
  * * *
  *
  *
  *

its first==biggest element is 5, but its length is 6, so it needs a 6*6 box
to just fit.

In the table below, each row-sum totals to PartitionsQ[n] by definition,
but its second half looks like 2*Reverse[PartitionsQ[n/2]].
This is {2,2,4,6,10,14,22,30,44,60,84,112,154 ..} in reverse.



Table[Count[Partitions[n],z_List/;Max[First[z],Length[z]]==m],{n,18},{m,n}]

{1}
{0, 2}
{0, 1, 2}
{0, 1, 2, 2}
{0, 0, 3, 2, 2}
{0, 0, 3, 4, 2, 2}
{0, 0, 2, 5, 4, 2, 2}
{0, 0, 1, 7, 6, 4, 2, 2}
{0, 0, 1, 6, 9, 6, 4, 2, 2}
{0, 0, 0, 7, 11, 10, 6, 4, 2, 2}
{0, 0, 0, 5, 14, 13, 10, 6, 4, 2, 2}
{0, 0, 0, 5, 15, 19, 14, 10, 6, 4, 2, 2}
{0, 0, 0, 3, 17, 22, 21, 14, 10, 6, 4, 2, 2}
{0, 0, 0, 2, 17, 29, 27, 22, 14, 10, 6, 4, 2, 2}
{0, 0, 0, 1, 17, 33, 36, 29, 22, 14, 10, 6, 4, 2, 2}
{0, 0, 0, 1, 15, 39, 45, 41, 30, 22, 14, 10, 6, 4, 2, 2}
{0, 0, 0, 0, 14, 41, 57, 52, 43, 30, 22, 14, 10, 6, 4, 2, 2}
{0, 0, 0, 0, 11, 47, 67, 69, 57, 44, 30, 22, 14, 10, 6, 4, 2, 2}

I am sorry, but the terms
1, 6, 9, 6, 4, 2 
do not match anything in the table

Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be



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