The coinvariant algebra, the quotient of the coordinate ring of affine space by the ideal generated by positive degree invariant polynomials, plays a basic role in algebraic combinatorics and the representation theory of the symmetric group, equipping its regular representation with a graded algebra structure. Using the coordinate ring of (P^1)^n in its Segre embedding, I will introduce a degeneration of the coinvariant algebra, the projective coinvariant algebra, which gives a bigraded structure on the regular representation of the symmetric group with interesting Frobenius character that generalises a classical result of Lusztig and Stanley. I will also show how this algebra contains bigraded versions of partial coinvariant algebras, coming from coordinate rings of all possible Segre embeddings. Hilbert functions of the bigraded algebras are controlled by natural expressions known from algebraic combinatorics.