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$k$-Regular Partitions

DEFINITION. A partition of an integer $n$ is said to be $k$-regular if none of the parts in the partition is a multiple of $k$.

For example, the unrestricted partitions of 5 are \[ 5,\quad 4+1, \quad 3+2, \quad 3+1 +1, \quad 2+ 2+1, \quad 2+1+1+1, \quad 1+1+1+1+1, \] while the 3-regular partitions of 5 are \[ 5,\quad 4+1, \quad 2+ 2+1, \quad 2+1+1+1, \quad 1+1+1+1+1. \]

Let $b_k(n)$ denote the number of $k$-regular partitions of $n$, so that, for example, $b_3(5)=5$.

The generating function for the sequence $\{b_k(n)\}_{n\geq 0}$ is \begin{equation}\label{bkngen} \sum_{n=0}^{\infty}b_k(n)q^n = \frac{\prod_{j=1}^{\infty}1-q^{kj}}{\prod_{j=1}^{\infty}1-q^{j}} = \frac{1}{\displaystyle{\prod_{j=1,\,k\nmid j}^{\infty}1-q^{j}}}. \end{equation} Thus the generating function for 3-regular partitions is \begin{multline*} \sum_{n=0}^{\infty}b_3(n)q^n = \frac{\prod_{j=1}^{\infty}1-q^{3j}}{\prod_{j=1}^{\infty}1-q^{j}} = \frac{1}{(1-q)(1-q^2)(1-q^4)(1-q^5)(1-q^7)\dots}\\ = 1 + q + 2 q^2 + 2 q^3 + 4 q^4 + 5 q^5 + 7 q^6 + 9 q^7 + 13 q^8 + 16 q^9 + 22 q^{10} +\dots. \end{multline*}

Hagis [1] found a Hardy–Ramanujan-Rademacher-type series for $b_k(n)$: \begin{multline*} b_k(n)=\frac{2\pi}{k\sqrt{24n+k-1}} \sum_{\stackrel{0 < d < \sqrt{k}}{d|k}} \sqrt{d(k-d^2)} \sum_{\stackrel{j\geq 1}{(j,k)=d}} \frac{1}{j} \\ \times \sum_{\stackrel{0\leq h < j}{(h,j)=1}} e^{-2\pi n h/j} \frac{\omega (h,j)}{\omega (kh/d,j/d)} I_1 \left ( \frac{\pi}{6j} \sqrt{ \frac{(24n+k-1)(k-d^2)}{k} } \right ), \end{multline*} where $I_1(z)$ is the Bessel function of the first kind, and $\omega(e,f)$ denotes $\exp(\pi i\,s(e,f))$, where the Dedekind sum $s(e,f)$ is defined by \[ s(e,f):=\sum_{r=1}^{f-1}\frac{r}{f}\left(\frac{er}{f}-\left \lfloor \frac{er}{f}\right \rfloor - \frac{1}{2}\right). \]

Just as partition congruences that hold for all integers in various arithmetic progressions have been found for $p(n)$, the counting function for unrestricted partitions, so also have similar congruences been found for $b_k(n)$, for various values of $k$. For example, the authors in [2] have shown that \[ b_4(9n+7)\equiv 0 (\mod 12), \] for all integers $n \geq 0$. Several similar results are known for other values of $k$.

[1] P. Hagis, Partitions with a restriction on the multiplicity of summands, Trans. Amer. Math. Soc. 155 (1971) 375–384.
[2] G. E. Andrews, M. D. Hirschhorn and J. A. Sellers, Arithmetic properties of partitions with even parts distinct, Ramanujan J. 23(1–3) (2010), 169–181.

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