  We study the combinatorics of addition using balanced digits, deriving an
analog of Holte's "amazing matrix" for carries in usual addition. The
eigenvalues of this matrix for base b balanced addition of n numbers are found
to be 1,1/b,...,1/b^n, and formulas are given for its left and right
eigenvectors. It is shown that the left eigenvectors can be identified with
hyperoctahedral Foulkes characters, and that the right eigenvectors can be
identified with hyperoctahedral Eulerian idempotents. We also examine the
carries that occur when a column of balanced digits is added, showing this
process to be determinantal. The transfer matrix method and a serendipitous
diagonalization are used to study this determinantal process.
