  The $n$-type vectors introduced by Geramita, Harima and Shin are in 1-1
correspondence with the Hilbert functions Artinian of lex ideals. Letting
$\mathbb{A} =\{a_1,..., a_n\}$ define the degrees of a regular sequence, we
construct ${\rm lpp}_{\le\mathbb{A}}$-vectors which are in 1-1 correspondence
with the Hilbert functions of certain lex plus powers ideals (depending on
$\mathbb{A}$). This construction enables us to show that the residual of a lex
plus powers ideal in an appropriate regular sequence is again a lex plus powers
ideal. We then use this result to show that the Eisenbud-Green-Harris
conjecture is equivalent to showing that lex plus powers ideals have the
largest last graded Betti numbers (it is well-known that the
Eisenbud-Green-Harris conjecture is equivalent to showing that lex plus powers
ideals have the largest first graded Betti numbers).
