A Frobenius symbol of a positive integer \(n\) is a two-rowed array of non-negative integers \begin{equation}\label{frobeq0} \left( \begin{matrix} a_1&a_2&a_3&\dots & a_r\\ b_1&b_2&b_3&\dots & b_r \end{matrix} \right) \end{equation} such that \begin{equation}\label{frobeq1} a_1>a_2>a_3>\dots >a_r\geq 0, \qquad b_1>b_2>b_3>\dots > b_r\geq 0, \end{equation} and \begin{equation}\label{frobeq1a} n=r+\sum_{i=1}^{r}a_i+\sum_{i=1}^{r}b_i. \end{equation}

The Frobenius symbol of a partition may be derived from its Ferrers diagram by
- deleting the main diagonal of the Durfee square;
- the remaining dots in the rows to the right of this diagonal are counted to give one row of the Frobenius symbol;
- the remaining dots in the columns below this diagonal are counted to give the second row of the Frobenius symbol.

This is illustrated in the following example for the partition \(7 + 7 + 6 + 6 + 5 + 5 + 4 + 3 + 3 + 2 + 1 + 1\) of 50:

Upon deleting the central diagonal of the Durfee square (red) and enumerating the remaining dots in the horizontal rows (green) and vertical columns (blue), the Frobenius symbol of the partition is formed by arranging these totals as two rows in an array: \[ \left( \begin{matrix} 6 & 5 & 3 & 2 & 0\\ 11 & 8 & 6 & 3 & 1 \end{matrix} \right). \] One of the reasons Frobenius devised this symbol was to create a notation that would make the conjugate of a partition immediately evident once the original partition was described. For example, it is clear that the Frobenius symbol for the partition conjugate to the one whose Ferrers diagram is in Figure 1. is derived by simply switching the two rows: \[ \left( \begin{matrix} 11 & 8 & 6 & 3 & 1\\ 6 & 5 & 3 & 2 & 0 \end{matrix} \right). \] If the strictly increasing conditions on the \(a_i,\, b_i\) are relaxed, the resulting object is called a Generalized Frobenius Partition or F-partition.

One example of a particular class of F-partitions that has been investigated in [1] (one of two classes considered in [1]) is the class where the conditions on the \(a_i,\, b_i\) is replaced with \begin{equation}\label{frobeq2} a_1 \geq a_2 \geq a_3 \geq \dots \geq a_r\geq 0, \qquad b_1 \geq b_2 \geq b_3 \geq \dots \geq b_r\geq 0, \end{equation} and up to \(k\) repetitions of an integer in any row is allowed.
Let \(\phi_k(n)\) denote the number of F-partitions \eqref{frobeq0}, with the entries satisfying \eqref{frobeq1} and \eqref{frobeq2}, and such that up to \(k\) repetitions of an integer in any row is allowed. Let the generating function for \(\{\phi_k(n)\}\) be defined, for \(|q|< 1\), by \begin{equation}\label{frobeq3} \Phi_k(q)=\sum_{n=0}^{\infty} \phi_k(n)q^n. \end{equation} It can easily be seen that \(\phi_1(n)\) is just the number of Frobenius symbols, and from the connection with partitions above, \(\phi_1(n)=p(n)\), where \(p(n)\) is the number unrestricted partitions of \(n\). Thus, from the page on generating functions for partitions, it is clear that \begin{equation}\label{frobeq4} \Phi_1(q)=\sum_{n=0}^{\infty} p(n)q^n =\frac{1}{(q;q)_{\infty}}. \end{equation}

Andrews derived many of the properties of \(\phi_k(n)\) and \( \Phi_1(q)\) in [1]. He showed that \begin{equation}\label{frobeq5} \Phi_k(q)=\frac{1}{(q;q)_{\infty}^k} \left ( \sum_{m_1,m_2,\dots , m_{k-1}=-\infty}^{\infty}\zeta^{R(m_1,m_2,\dots , m_{k-1})} q^{Q(m_1,m_2,\dots , m_{k-1})} \right ), \end{equation} where \(\zeta = \exp(2\pi i/(k+1))\), and \begin{align*} R(m_1,m_2,\dots , m_{k-1})&= (k-1)m_1+(k-2)m_2+\dots + m_{k-1},\\ Q(m_1,m_2,\dots , m_{k-1})&=\sum_{i=1}^{k-1}m_i^2 + \sum_{1\leq i < j \leq k-1}m_im_j. \end{align*} Closed forms were given for \(\Phi_k(q)\) in the cases \(k=2\) and \(k=3\). For \(k=2\) this takes the forms \[ \Phi_2(q)=\frac{1}{(q;q)_{\infty}(q^2,q^3,q^9,q^{10};q^{12})_{\infty}}. \] Andrews also proved a number of congruence properties for some F-partition functions, similar to Ramanujan's partition congruences for the ordinary partition function. For example, he showed that \[ \phi_2(5n+3) \equiv 0 (\mod 5), \,\,\forall n \geq 0. \] F-partitions continue to be an objective of ongoing research, and many more of their properties have been developed since the memoir [1] of Andrews.

[1] G.E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 49 (1984), no. 301, iv+44.

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