  We compute the equivariant partition function of the six-dimensional M-string
SCFTs on a background with the topology of a product of a two-dimensional torus
and an ALE singularity. We determine the result by exploiting BPS strings
probing the singularity, whose worldvolume theories we determine via a chain of
string dualities. A distinguished feature we observe is that for this class of
background the BPS strings' worldsheet theories become relative field theories
that are sensitive to finer discrete data generalizing to 6d the familiar
choices of flat connections at infinity for instantons on ALE spaces. We test
our proposal against a conjectural 6d N = (1,0) generalization of the Nekrasov
master formula, as well as against known results on ALE partition functions in
four dimensions.
