# Article about partitions with forbidden parts/multiplicities

I am looking for an article, most likely from the 90s, that generalized the bijection between partitions with odd and distinct parts by explaining how a bijection between the forbidden parts could be transformed into a bijection of the partitions.

(So, in the example above, a bijection can be formed using the fact that $(2k)$ is in bijection with $(k,k)$ which form the forbidden parts on both sides.)

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Possible articles you might have seen could be

M.V. Subbarao, Partition theorems for Euler pairs, Proc. Amer. Math. Soc. 28 (1971), no. 2, 330-336.

Here the author characterizes all triples $(A,B,r)$ so that the number of partitions with parts in $A$ is equal to the number of partitions with parts in $B$ with no part repeated more than $r-1$ times. These are the sets satisfying $rB\subset B$ and $A=B-rB$.

J.B. Remmel, Bijective Proofs of Some Classical Partition Identities, J. Combin. Theory Ser. A 33 (1982), 273–286

This paper proves a very general statement phrased in terms of forbidden patterns. Namely given two sequences of non-empty multisets $\mathcal A=\lbrace A_i\rbrace _{i\in \omega}$ and $\mathcal B=\lbrace B_i\rbrace _{i\in \omega}$, let $|M|$ denote the sum of the elements in $M$ as a multiset. Then if $$|\bigcup _{i\in S} A_i|=|\bigcup _{i\in S} B_i|$$ holds for all $S\subset \omega$, the number of partitions with no $\mathcal A$ patterns is equal to the number of partitions with no $\mathcal B$ patterns.

K.M. O’Hara, Bijections for Partition Identities, J. Combin. Theory Ser. A 49 (1988), 13–25.

While Remmel's approach gives a bijection based on the Garsia-Milne involution, this paper shows that the algorithm for producing bijections can be made considerably faster at least in the case of disjoint multisets.

On the other hand, it could have been a more recent article which most probably references at least one of these papers. You can find an extended list of references as well as various statements which generalize the bijection you mention, in Igor Pak's survey on partition bijections, the relevant section being section 8 (survey is also available from his website). This is also the topic of Herbert S. Wilf's notes "Lectures on Integer Partitions" (which you can find here).

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Thank you, the last two articles are exactly what I was looking for. I appreciate your useful summaries that makes me recognize them immediately. I also appreciate your other links. – user11235 Nov 7 '11 at 20:07