Gap-frequency partitions, or g-f partitions, were introduced by George Andrews in [1] to provide a combinatorial explanation of the three Rogers-Selberg identities: \begin{align} \sum_{n=0}^\infty \frac{ q^{2n(n+1)} } {(q^2;q^2)_n (-q;q)_{2n+1}} &= \frac{ (q,q^6,q^7;q^7)_{\infty} }{ (q^2;q^2)_{\infty} }; \\ \sum_{n=0}^\infty \frac{ q^{2n(n+1)} } {(q^2;q^2)_n (-q;q)_{2n}} &= \frac{ (q^2,q^5,q^7;q^7)_{\infty} }{ (q^2;q^2)_{\infty} }; \\ \sum_{n=0}^\infty \frac{ q^{2n^2} } {(q^2;q^2)_n (-q;q)_{2n}} &= \frac{ (q^3,q^4,q^7;q^7)_{\infty} }{ (q^2;q^2)_{\infty} } . \end{align} DEFINITION. A partition \(\pi\) is said to be a gap-frequency (or g-f) partition if whenever a summand \(s\) appears exactly \(t\) times, the next larger part is at least \(s + t\), and if it is exactly \(s + t\) it can appear at most \(t\) times.

Example. 1 + 4 + 4 + 4 + 7 + 7 + 7 is a g-f partition. Neither 1 + 4 + 4 + 4 + 7 + 7 + 7 + 7 nor 1 + 4 + 4 + 4 + 6+ 6 + 6 is a g-f partition.

Gap-frequency partitions were further studied by David Bressoud in [2].

DEFINITION. For positive integers \( r, x\) and \(n\), let \(S_{r,x} (n)\) denote the number of g-f partitions of \(n\) in which no part appears more than \(x\) times and one appears at most \(r — 1\) times.

Bressoud proved the following theorem in [2].

Theorem For positive integers \(r, x\) and n and for \(|q| < 1\), let \[ M(m_l,\dots , m_x ) = M = \sum_{j=1}^x j^2 \binom{m_j}{2} + \sum_{1\leq i < j \leq x}ijm_im_j. \] Then \begin{equation*} \sum_{m_1, \dots , m_x\geq 0} \frac{q^{M+m_1+2m_2+\dots + (r-1)m_{r-1}+2(rm_r+(r+1)m_{r+1}+\dots + x m_x)}} {(q;q)_{m_1}(q^2;q^2)_{m_2}\dots (q^x;q^x)_{m_x}}= \sum_{n=0}^{\infty}S_{r,x}(n)q^n. \end{equation*} Bressoud also generalized the concept of g-f partitions.

DEFINITION. A partition \(\pi\) is said to be a k-fold g-f partition if whenever a summand \(s\) appears exactly \(t\) times, the next larger part is at least \(s + kt\), and if it is exactly \(s + kt\) it can appear at most \(t\) times.

DEFINITION. For positive integers \(r, x, k\) and \(n\), let \(S_{r,x,k} (n)\) denote the number of \(k\)-fold g-f partitions of \(n\) in which no part appears more than \(x\) times and one appears at most \(r — 1\) times.

Bressoud noted that by an argument similar to that which he used to prove the theorem above about the generating function for \(\{S_{r,x}(n)\}_{n\geq 0}\), it can be shown that \begin{equation*} \sum_{m_1, \dots , m_x\geq 0} \frac{q^{kM+m_1+2m_2+\dots + (r-1)m_{r-1}+2(rm_r+(r+1)m_{r+1}+\dots + x m_x)}} {(q;q)_{m_1}(q^2;q^2)_{m_2}\dots (q^x;q^x)_{m_x}}= \sum_{n=0}^{\infty}S_{r,x,k}(n)q^n. \end{equation*}

[1] G. E. Andrews, Gap-frequency partitions and the Rogers-Selberg identities. Ars Combin. 9 (1980), 201–210.
[2] D. M. Bressoud, A note on gap-frequency partitions. Pacific J. Math. 89 (1980), no. 1, 1–6.

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