  G. E. Murphy showed in 1983 that the centre of every symmetric group algebra
has an integral basis consisting of a specific set of monomial symmetric
polynomials in the Jucys--Murphy elements. While we have shown in earlier work
that the centre of the group algebra of S_3 has exactly three additional such
bases, we show in this paper that the centre of the group algebra of S_4 has
infinitely many bases consisting of monomial symmetric polynomials in
Jucys--Murphy elements, which we characterize completely. The proof of this
result involves establishing closed forms for coefficients of class sums in the
monomial symmetric polynomials in Jucys--Murphy elements, and solving several
resulting exponential Diophantine equations with the aid of a computer.
  Our initial motivation was in finding integral bases for the centre of the
Iwahori--Hecke algebra, and we address this question also, by finding several
integral bases of monomial symmetric polynomials in Jucys--Murphy elements for
the centre of the Iwahori--Hecke algebra of S_4.
