### Basic Examples

Basic Examples

The largest part is 4 and there are 3 parts; :

4-3=1

1

The largest part and the number of parts are both 5; :

5-5=0

In[]:=

Out[]=

0

### Neat Examples

Neat Examples

Ramanujan discovered and proved analytically that the number of partitions of is divisible by 5:

5m+4

In[]:=

PartitionsP[Range[4,44,5]]

Out[]=

{5,30,135,490,1575,4565,12310,31185,75175}

Dyson conjectured and Atkins–Swinnerton–Dyer proved that the partitions of a number of the form can be split into five sets of equal size according to the rank mod 5:

5m+4

In[]:=

Length/@GatherByIntegerPartitions[9],Mod@#,5&

Out[]=

{6,6,6,6,6}

Here is another example:

In[]:=

Length/@GatherByIntegerPartitions[24],Mod@#,5&

Out[]=

{315,315,315,315,315}

A similar result holds for 7; in this case the partitions of a number of the form can be split into 7 sets of equal size, according to the rank mod 7:

7m+5

In[]:=

Length/@GatherByIntegerPartitions[12],Mod@#,7&

Out[]=

{11,11,11,11,11,11,11}