We explain how to compute top-dimensional intersections of
ψ
\psi
-classes on
M
¯
1
,
n
(
m
)
\overline {M}_{1,n}(m)
, the moduli space of
m
m
-stable curves. On the spaces
M
¯
1
,
n
\overline {M}_{1,n}
, these intersection numbers are determined by two recursions, namely, the string equation and dilaton equation. We establish, for each fixed
m
≥
1
m \geq 1
, an analogous pair of recursions that determine these intersection numbers on the spaces
M
¯
1
,
n
(
m
)
\overline {M}_{1,n}(m)
.