  Let the sign of a standard Young tableau be the sign of the permutation you
get by reading it row by row from left to right, like a book. A conjecture by
Richard Stanley says that the sum of the signs of all SYTs with n squares is
2^[n/2]. We present a stronger theorem with a purely combinatorial proof using
the Robinson-Schensted correspondence and a new concept called chess tableaux.
  We also prove a sharpening of another conjecture by Stanley concerning
weighted sums of squares of sign-imbalances. The proof is built on a remarkably
simple relation between the sign of a permutation and the signs of its
RS-corresponding tableaux.
