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 SmallestPartsFunction

Source Notebook

Total number of smallest parts in the partitions of a positive integer

Contributed by: George Beck

ResourceFunction["SmallestPartsFunction"][n]

gives the total number of smallest parts in the integer partitions of the positive integer n.

Details and Options

A partition of n is a list of weakly decreasing positive integers that add up to n. For instance, written compactly, these are the five partitions of 4: 4, 31, 22, 211, 1111.
ResourceFunction["SmallestPartsFunction"][n] returns the Total of all the minimal elements in all the partitions of n.

Examples

Basic Examples (2) 

Here is an illustrative example. Start with the integer partitions of 3:

In[1]:=
IntegerPartitions@3
Out[1]=

Here are their smallest parts:

In[2]:=
Min /@ IntegerPartitions@3
Out[2]=

Compute the number of smallest parts per partition:

In[3]:=
Count[#, Min[#]] & /@ IntegerPartitions@3
Out[3]=

Find their total:

In[4]:=
Total@%
Out[4]=

This is precisely the value returned by SmallestPartsFunction:

In[5]:=
ResourceFunction["SmallestPartsFunction"][3]
Out[5]=

Here are the first 10 numbers in the smallest part function (denoted spt) sequence (OEIS A092269):

In[6]:=
ResourceFunction["SmallestPartsFunction"] /@ Range@10
Out[6]=

Neat Examples (3) 

The function spt(n) satisfies some congruences similar to Ramanujan’s congruences for the partition function p(n).

Numbers of the form 5n+4 have spt divisible by 5:

In[7]:=
Table[ResourceFunction["SmallestPartsFunction"][5 n + 4], {n, 10}]
Out[7]=
In[8]:=
Mod[#, 5] == 0 & /@ %
Out[8]=

Numbers of the form 7n+5 have spt divisible by 7:

In[9]:=
Table[ResourceFunction["SmallestPartsFunction"][7 n + 5], {n, 7}]
Out[9]=
In[10]:=
Mod[#, 7] == 0 & /@ %
Out[10]=

Numbers of the form 13n+6 have spt divisible by 13:

In[11]:=
Table[ResourceFunction["SmallestPartsFunction"][13 n + 6], {n, 4}]
Out[11]=
In[12]:=
Mod[#, 13] == 0 & /@ %
Out[12]=

Publisher

George Beck

Version History

  • 1.0.0 – 31 May 2019

Related Resources

License Information