Partitions with initial repetitions were introduced by George Andrews in [1], and were defined as follows.

DEFINITION. Let $k$ be a positive integer. A partition with initial $k$-repetitions is a partition in which if the part $j$ appears at least $k$ times, then all positive integers less than $j$ appear as parts at least $k$ times.

Andrews proved several results about these types of partitions.

Theorem 1 (Andrews, [1]). The number of partitions of $n$ with initial $k$-repetitions equals the number of partitions of $n$ into parts not divisible by $2k$ and also equals the number of partitions of $n$ in which no part is repeated more than $2k − 1$ times.

The proof of Andrews is short, using generating functions, $q$-series/product manipulations, and a special case of the $q$-binomial theorem. The proof is included here for completeness. The generating function for partitions with initial $k$-repetitions is \begin{align*} &\sum_{n=0}^{\infty}\frac{q^{k.1+k.2+\dots +k.n}}{(q;q)_n}\prod_{j=n+1}^{\infty}(1+q^j+q^{2j}+\dots +q^{(k-1)j})\\ &=\sum_{n=0}^{\infty}\frac{q^{kn(n+1)/2}}{(q;q)_n}\prod_{j=n+1}^{\infty}\frac{1-q^{kj}}{1-q^j}\\ &=\frac{(q^k;q^k)_{\infty}}{(q;q)_{\infty}}\sum_{n=0}^{\infty}\frac{q^{kn(n+1)/2}}{(q^k;q^k)_n}\\ &=\frac{(q^k;q^k)_{\infty}}{(q;q)_{\infty}}(-q^k;q^k)_{\infty}\\ &=\frac{(q^{2k};q^{2k})_{\infty}}{(q;q)_{\infty}}\\ &=\prod_{j=1}^{\infty}(1+q^j+q^{2j}+\dots +q^{(2k-1)j}). \end{align*} The next-t0-last infinite product is the generating function for partitions into parts not divisible by $2k$, and the last infinite product is the generating function for partitions in which no part is repeated more than $2k − 1$ times.

Andrews in [1] also considered partitions in which the repeating initial parts have the same parity, and proved some results about these.

DEFINITION. Let $F_e(n)$ denote the number of partitions of $n$ in which no odd parts are repeated and if an even part $2j$ is repeated then each even positive integer smaller than $2j$ appears in the partition as a repeated part and no odd integers smaller than $2j$ appear.

Theorem 2 (Andrews, [1]). $F_e(n)$ equals the number of partitions of $n$ into parts $\not \equiv 0,\pm 2 (\mod 7)$.
The proof uses one of the Rogers-Selberg identities.

A bijective proof of Theorem 2 was given in [2] by William J. Keith.

DEFINITION. Let $F_o(n)$ denote the number of partitions of $n$ in which no even parts are repeated and no even part is smaller than a repeated odd part, and if an odd part $2j −1$ is repeated then each odd positive integer smaller than $2j − 1$ appears in the partition as a repeated part.

Theorem 3 (Andrews, [1]). $ \sum_{n=1}^{\infty} F_o(n)q^n = (−q;q)_{\infty}f(q^2), $ where $f(q)$ is one of Ramanujan’s seventh order mock theta functions, defined by \[ f(q) = \sum_{n=1}^{\infty} \frac{q^{n^2}} {(q^n;q)_n}. \] Partitions in which the initial repetitions are multiples of 3 were also considered in [1].

DEFINITION. $B(n)$ denotes the number of partitions of $n$ wherein:
(1) all parts are $> 1$,
(2) if a multiple of 3, say $3j$, appears more than once, then each positive multiple of 3 smaller than $3j$ appears more than once as a part, no non-multiples of 3 smaller than $3j + 2$ appear,
(3) no two parts that are non-multiples of 3 differ by exactly 1.

Theorem 4 (Andrews [1]). $B(n)$ equals the number of partitions of $n$ into parts $\not \equiv 0,\pm 1 (\mod 9)$.

[1] G. E. Andrews, Partitions with initial repetitions, Acta Math. Sinica, English Series 25(9) (2009), 1437–1442.

[2] W. J. Keith, A Bijection for Partitions with initial repetitions. The Ramanujan Journal, February 2012, Volume 27, Issue 2, pp 163-167.

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