Algebraic basis of the algebra of block-symmetric polynomials on $\ell_1 \oplus \ell_{\infty}$

We consider so called block-symmetric polynomials on sequence spaces $\ell_1\oplus \ell_{\infty}, \ell_1\oplus c, \ell_1\oplus c_0,$ that is, polynomials which are symmetric with respect to permutations of elements of the sequences. It is proved that every continuous block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ can be uniquely represented as an algebraic combination of some special block-symmetric polynomials, which form an algebraic basis. It is interesting to note that the algebra of block-symmetric polynomials is infinite-generated while $\ell_{\infty}$ admits no symmetric polynomials. Algebraic bases of the algebras of block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ and $\ell_1\oplus c_0$ are described.