The functions p(n), p_k(n), and the standard restricted partition families

For integers $n$ and $k$, define:

• for $n>0$, a partition of $n$ is one from Partitions of a positive integer, while for $n=0$ there is exactly one partition, the empty partition ${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}:=()$;
• $p(n)$ is the number of partitions of $n$ when $n\ge 0$, with $p(0)=1$, and $p(n)=0$ for $n<0$;
• $p_k(n)$ is the number of partitions of $n$ into exactly $k$ positive parts when $n\ge 0$ and $k\ge 0$, and $p_k(n)=0$ if $n<0$ or $k<0$.

Thus $p_0(0)=1$, while $p_0(n)=0$ for $n\mathrm{\ne }0$.

For the empty partition, the number of parts is $0$ and the largest part is defined to be $0$. The empty partition is regarded as having both distinct parts and odd parts.

For a nonempty partition $\lambda =({\lambda }_1,\dots ,{\lambda }_r)$ of $n$:

• $\lambda$ has at most $k$ parts when $r\le k$;
• $\lambda$ has largest part $k$ when ${\lambda }_1=k$;
• $\lambda$ has parts at most $k$ when ${\lambda }_1\le k$;
• $\lambda$ has distinct parts when ${\lambda }_1>{\lambda }_2>\cdots >{\lambda }_r$;
• $\lambda$ has odd parts when every ${\lambda }_i$ is odd.

Write $p_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}(n)$ for the number of partitions of $n$ into distinct parts and $p_{\mathrm{o}\mathrm{d}\mathrm{d}}(n)$ for the number of partitions of $n$ into odd parts.