2013/09/19 by Persi Diaconis, Jason Fulman, Diaconis, Persi +1
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR
paper · pdf · doi:10.48550/arxiv.1309.5116
18 pages
arxiv created 2013/09/19 · openalex publication_date 2013/09/19 · arxiv updated 2013/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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/bn, 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.