  We give a new presentation for the partition algebras. This presentation was
discovered in the course of establishing an inductive formula for the partition
algebra Jucys-Murphy elements defined by Halverson and Ram [European J. Combin.
26 (2005), 869-921]. Using Schur-Weyl duality we show that our recursive
formula and the original definition of Jucys-Murphy elements given by Halverson
and Ram are equivalent. The new presentation and inductive formula for the
partition algebra Jucys-Murphy elements given in this paper are used to
construct the seminormal representations for the partition algebras in a
separate paper.
