  We study the lattice of closed ideals in the algebra of continuous linear
operators acting on $p$th Tandori and $p'$th Ces\`{a}ro sequence spaces,
$1\leqslant p<\infty$, which we show are isomorphic to the classical sequence
spaces $(\oplus_{n=1}^\infty\ell_\infty^n)_p$ and
$(\oplus_{n=1}^\infty\ell_1^n)_{p'}$, respectively. We also show that Tandori
sequence spaces are complemented in certain Lorentz sequence spaces, and that
the lattice of closed ideals for certain other Lorentz and Garling sequence
spaces has infinite cardinality.
