Partition

Partition into k

$p_k(n)$ is number of partitions of n into exactly k parts. $p_k(n)=p_{k-1}(n-1)+p_k(n-k)$

\eqalign{ &1\cr &1 \ 1\cr &1 \ 1 \ 1 \cr &1 \ 2 \ 1 \ 1 \cr &1 \ 2 \ 2 \ 1 \ 1 \cr &1 \ 3 \ 3 \ 2 \ 1 \ 1 \cr &1 \ 3 \ 4 \ 3 \ 2 \ 1 \ 1 \cr &1 \ 4 \ 5 \ 5 \ 3 \ 2 \ 1 \ 1 \cr &1 \ 4 \ 7 \ 6 \ 5 \ 3 \ 2 \ 1 \ 1 \cr }

https://en.wikipedia.org/wiki/Triangle_of_partition_numbers

Partition

• A partition of a positive integer $n$, (also called an integer partition), is a way of writing $n$ as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition.

The partition function $p(n)$ equals the number of possible partitions of a non-negative integer $n$.

The values of this function for $0,1,2,3,\dots$ are: (A000041) $1,1,2,3,5,7,11,15,22,30,42,56,\dots$

$p(n)=\sum _{k=0}^n{p}_k(n)$

https://en.wikipedia.org/wiki/Partition_(number_theory)

Partition with distinct parts

$Q(n)$, also denoted $q(n)$, gives the number of ways of writing the integer $n$ as a sum of positive integers without regard to order with the constraint that all integers in a given partition are distinct. (A000009) $1,1,2,2,3,4,5,6,8,10,\dots$

https://en.wikipedia.org/wiki/Partition_(number_theory)#Odd_parts_and_distinct_parts