Let $S$ denote a class of partitions (for example unrestricted partitions, partitions into distinct parts, partitions into distinct parts with gap at least 2 between consecutive parts, etc., etc.), and for a given positive integer $n$ let $p_S(n)$ denote the number of partitions of $n$ in $S$ ($p_S(0)$ is taken to have value 1). As usual, if $\pi$ denotes a partition of $n$, then $|\pi|$ is used to denote the sum of the parts of $\pi$, which of course then means that $|\pi|=n$.

If $\pi$ is used to denote an arbitrary partition in $S$ (without specifying which integer it is a partition of), then \begin{equation}\label{weighteq1} \sum_{\pi \in S}q^{|\pi|}= \sum_{\pi \in S}1\cdot q^{|\pi|}= \sum_{n=0}^{\infty}p_S(n)q^n \end{equation} is the ordinary generating function for the sequence $\{p_S(n)\}_{n=0}^{\infty}$. Note that the last equality follows since \begin{equation}\label{weighteq2} \sum_{\pi \in S, |\pi|=n}1= p_S(n). \end{equation}
In [A1], Alladi considered the effect of replacing the "1" multiplying the $q^{|\pi|}$ in the middle sum at \eqref{weighteq1} by a "weight" $\omega_S(\pi)$. In particular he started the investigation of the following general problem:
If $S$ and $T$ are classes of partitions such that $S\subset T$, how to find some naturally defined weight function $\omega_S(\pi)$ such that \begin{equation}\label{weighteq3} \sum_{\pi \in S}\omega_S(\pi)\cdot q^{|\pi|}= \sum_{n=0}^{\infty}p_T(n)q^n, \end{equation} or alternatively, such that \begin{equation}\label{weighteq4} \sum_{\pi \in S, |\pi|=n}\omega_S(\pi)= p_T(n). \end{equation} Alladi [A1] considered four classes of partitions and their respective counting functions:
(i) $p(n) =$ the number of unrestricted partitions of $n$,
(ii) $Q(n) =$ the number of partitions of $n$ into distinct parts,
(iii) $g(n) =$ the number of partitions of $n$ in which the even parts do not repeat,
and
(iv) $\rho(n) =$ the number of partitions of $n$ with difference $\geq 2$ between parts.

Alladi's first result connects partitions counted by $p(n)$ (for later use, denote the set of all such partitions by $U$) and those counted by $\rho(n)$ (denote the set of all partitions of this type by $R$, since the ordinary generating function function for this partitions is given by series side of one of the Rogers-Ramanujan identities).

Supose that $\pi$ is a partition in $R$, with \[ |\pi|=h_1+h_2+\dots + h_{\nu}, \qquad h_i-h_{i+1}\geq 2, \qquad i=1, 2, \dots \nu -1. \] Define $h_{\nu +1} =-1$ and then define the weight function \[ \omega_R (\pi) = \prod_{i=1}^{\nu}(h_i-h_{i+1}-1). \] THEOREM 1 (Alladi [A1]). Let $R$ denote the set of all partitions with minimal difference 2. Then \begin{equation}\label{weighteq5} p(n) = \sum_{\pi \in R,\, |\pi|=n}\omega_R(\pi). \end{equation} A primary partition of an integer $n$ is a partition whose Ferrers graph contains no nodes below the Durfee square (the series side of the the Rogers-Ramanujan identity alluded to above may also be considered as the generating function for this type of partition). Let $D_1$ denote the set of all such partitions, and for a partition $\pi \in D_1$ with \[ |\pi|=b_1+b_2+\dots + b_{\nu}, \qquad b_1\geq b_2\geq \dots \geq b_{\nu}>0, \] define $b_{\nu +1} =\nu$ and then define the weight function \[ \omega_1 (\pi) = \prod_{i=1}^{\nu}(b_i-b_{i+1}+1). \] THEOREM 2 (Alladi [A1]). Let $D_1$ denote the set of all partitions each of whose parts is at least as large as the number of parts. Then \begin{equation}\label{weighteq6} p(n) = \sum_{\pi \in D_1,\, |\pi|=n}\omega_1(\pi). \end{equation} Several other theorems of a similar type and varying degrees of complexity were also proved by Alladi in the same paper [A1], one of which is described next.

Let $D$ denote the set of partitions into distinct parts, , and for a partition $\pi \in D$ with \[ |\pi|=b_1+b_2+\dots + b_{\nu}, \qquad b_1> b_2> \dots > b_{\nu}>0, \] define $b_{\nu +1} =0$ and then define the weight function \[ \omega (\pi) = 2^k, \] if there are there are exactly $k$ gaps $b_i − b_{i+1} \geq 2$. For a positive integer $n$, let $Q(n;k)$ denote the number of such partitions of $n$. Recall from above that $g(n) =$ the number of partitions of $n$ in which the even parts do not repeat.

THEOREM 3 (Alladi [A1], slightly reworded). Let $D$ denote the set of all partitions into distinct parts, and for $\pi \in D$, let $\omega (\pi)$ and $Q(n;k)$ be as defined above. Then \begin{equation}\label{weighteq7} g(n) = \sum_{k}Q(n;k)2^k= \sum_{\pi \in D,\, |\pi|=n}\omega(\pi). \end{equation} Similar weighted partition identities were derived subsequently, both by Alladi and others.

[A1] K. Alladi, Partition identities involving gaps and weights, Trans. Amer. Math. Soc. 349 (1997), no. 12, 5001-5019.

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