The subject of the paper under review is the study of so-called generalized Frobenius partitions (or simply F-partitions) in \(k\)-colors as introduced by G. E. Andrews [Generalized Frobenius partitions. Providence, RI: American Mathematical Society (AMS) (1984; Zbl 0544.10010)], Sect. 4. These are F-partitions of the form \[ \begin{pmatrix}a_1 & a_2 & \dots & a_m \\ b_1 & b_2 & \dots & b_m \end{pmatrix}, \] where an integer may appear at most \(k\) times in each row and the order of the integers \(a_i\), \(b_i\) is assumed to be non-increasing. Such an F-partition is associated to the integer \[ n = m + \sum_{i=1}^m(a_i+b_i). \] The authors study the numbers \(c\psi_k(n)\) of all such F-partitions assigned to an integer \(n\) and more generally the numbers \(c\psi_{k,a}(n)\) of \((k,a)\)-colored F-partitions of \(n\), where \(a\in \mathbb{Z}+\frac{k}{2}\) is assumed to be non-negative.
A well known method to gain information on combinatorial quantities is the investigation the analytic properties of their associated generating function. For the case of the partition function and generalizations thereof, it is an established fact (see, e.g., [G. E. Andrews, Generalized Frobenius partitions. Providence, RI: American Mathematical Society (AMS) (1984; Zbl 0544.10010)]) that the generating function can essentially be expressed in terms of elliptic modular forms. The present paper adds new insight in this regard as it manages to relate the generating function \[ C\Psi_{k,a}(q) = \sum_{n=0}^\infty c\psi_{k,a}(n)q^n \] to a Jacobi form. More precisely, the following theorem is proved:
Let \(\vartheta(z,\tau) = \sum_{n\in \mathbb{Z}+\frac{1}{2}}e^{\pi in^2\tau+2\pi in(z+\frac{1}{2})}\) be the usual Jacobi theta function. For \(k\in \mathbb{Z}_{>0}\) we define a Jacobi form by \[ F_k(z;\tau):= \left(\frac{-\vartheta(z+\frac{1}{2},\tau)}{q^{\frac{1}{12}}\eta(\tau)}\right)^k, \] where \(\eta\) is the Dedekind eta function. Moreover, let \(\zeta = e^{2\pi i z}\). Then the \(\zeta^a\)-coefficient of \(F_k(z;\tau)\) is given by the generating function \(C\Psi_{k,a}(q)\).
Subsequently, under the assumption that \(k=2l\) is even, the theta decomposition (see [M. Eichler and D. Zagier, The theory of Jacobi forms. Boston-Basel-Stuttgart: Birkhäuser (1985; Zbl 0554.10018)], Chap. II, \(\S\) 5, (5), for details) \[ F_k(z,\tau) = \sum_{ a \bmod{2l}} H_{l,a}(\tau)\vartheta_{l,a}(z,\tau) \] of \(F_k(z,\tau)\) is exploited to derive a recursive formula for the functions \(H_{l,a}(\tau)\). As a consequence, the authors provide an algorithm to compute \(C\Psi_{k,a}(q)\) for any \(k\) and \(a\).
In Section 5, the previously obtained results are used to proof some congruence relations for the quantities \(c\psi_{k,a}(n)\). Finally, Section 6 of the paper presents a relation between shifted Motzkin paths (certain lattice paths) and \(C\Psi_{k,a}(q)\). Thereby, a connection between Motzkin paths and \((k,a)\)-colored F-partitions is established. These results extend the work of B. Drake [Discrete Math. 309, No. 12, 3936–3953 (2009; Zbl 1228.05032)].