Let \(S_l\) and \(S_2\) be two subsets of the natural numbers \(\mathbb{N}\). The pair \((S_1, S_2)\) is called an Euler-pair if for all natural numbers \(n\), the number of partitions of \(n\) into distinct parts taken from \(S_1\) equals the number of partitions of \(n\) into parts taken from \(S_2\) .

As Andrews noted in [1], Euler's theorem that the partitions of a natural number \(n\) into distinct parts are equinumerous with the partitions of \(n\) into odd parts is a particular case of this theorem, with \[ S_1=\{1,2,3,\dots \}, \qquad S_2=\{1,3,5,\dots \}. \] Andrews also observed that Euler's theorem that every positive integer \(n\) has a unique representation as a sum of distinct powers of 2, is also a special case of this general theorem, with \[ S_1=\{1,2,4,8,\dots \}, \qquad S_2=\{1\}. \] An example of an Euler pair given by I.J. Schur is \[ S_1=\{m\in \mathbb{N}|m\not \equiv 0 \mod 3 \}, \qquad S_2=\{m\in \mathbb{N}|m \equiv 1,5 \mod 6 \}, \] and an example given by H. Göllnitz is \[ S_1=\{m\in \mathbb{N}|m \equiv 2,4,5 \mod 6 \}, \qquad S_2=\{m\in \mathbb{N}|m \equiv 2,5,11 \mod 12 \}. \] Let \(mS\) denotes the set \[ mS=\{mn|n\in S\}, \] and let \[ S_i-S_j=\{n|n\in S_i, n \not \in S_j\}. \] Andrews proved the following theorem in [1]:

Theorem. The pair \((S_1, S_2)\) is an Euler pair if and only if \(2S_1\subseteq S_1\) and \(S_2=S_1-2S_1\).

The proof follows from considering the corresponding generating functions for partitions from the sets \(S_l\) and \(S_2\).

[1] G.E Andrews Two theorems of Euler and a general partition theorem Proc. Amer. Math. Soc., 20 (1969), pp. 499-502.

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