§26.10 Integer Partitions: Other Restrictions

$p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. $p(𝒟k,n)$ denotes the number of partitions of $n$ into parts with difference at least $k$. $p({𝒟}^{\prime }3,n)$ denotes the number of partitions of $n$ into parts with difference at least 3, except that multiples of 3 must differ by at least 6. $p(𝒪,n)$ denotes the number of partitions of $n$ into odd parts. $p(\in S,n)$ denotes the number of partitions of $n$ into parts taken from the set $S$. The set $\{n\ge 1|n\equiv \pm j(modk)\}$ is denoted by $A_{j,k}$. The set $\{2,3,4,\mathrm{\dots }\}$ is denoted by $T$. If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. See Table 26.10.1.

Throughout this subsection it is assumed that .

where the last right-hand side is the sum over $m\ge 0$ of the generating functions for partitions into distinct parts with largest part equal to $m$.

where the inner sum is the sum of all positive odd divisors of $t$.

where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$.

where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$.

In exact analogy with (26.9.8), we have

where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$.

where the inner sum is the sum of all positive divisors of $t$ that are in $S$.

Equations (26.10.13) and (26.10.14) are the Rogers–Ramanujan identities. See also §17.2(vi).

Note that $p({𝒟}^{\prime }3,n)\le p(𝒟3,n)$, with strict inequality for $n\ge 9$. It is known that for $k>3$, $p(𝒟k,n)\ge p(\in A_{1,k+3},n)$, with strict inequality for $n$ sufficiently large, provided that $k=2^m-1,m=3,4,5$, or $k\ge 32$; see Yee (2004).

where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and

with

and

The quantity $A_k(n)$ is real-valued.