  For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that
the r-variation of the
  Walsh-Fourier sums are finite, for r sufficiently large as function of w.
(That r is a function of w is necessary.) This strengthens a result of
Hunt-Young and is a weighted extension of a variation norm Carleson theorem of
Oberlin-Seeger-Tao-Thiele-Wright. The proof uses phase plane analysis and a
weighted extension of a variational inequality of Lepingle.
