  The question of which functions acting entrywise preserve positive
semidefiniteness has a long history, beginning with the Schur product theorem
[Crelle 1911], which implies that absolutely monotonic functions (i.e., power
series with nonnegative coefficients) preserve positivity on matrices of all
dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942,
1959] shows the converse: there are no other such functions.
  Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math.
Soc. 2015] classified the entrywise positivity preservers in all dimensions,
which act only on the off-diagonal entries. These two results are at "opposite
ends", and in both cases the preservers have to be absolutely monotonic.
  We complete the classification of positivity preservers that act entrywise
except on specified "diagonal/principal blocks", in every case other than the
two above. (In fact we achieve this in a more general framework.) This yields
the first examples of dimension-free entrywise positivity preservers - with
certain forbidden principal blocks - that are not absolutely monotonic.
