Combinatorics of balanced carries

Combinatorics of Balanced Carries

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, and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes c...

Carries, Group Theory, and Additive Combinatorics

When numbers are added in the usual way carries occur along the route. These carries cause a mess and it is natural to seek ways to minimize them. This paper proves that balanced arithmetic minimizes the proportion of carries. It also positions carries as cocycles in group theory and shows that if coset representatives for a finite-index normal subgroup H in a group G can be chosen so that the ...

The Order Steps of an Analytic Combinatorics

‎Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures‎. ‎This theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines‎, ‎including probability theory‎, ‎statistical physics‎, ‎computational biology and information theory‎. ‎With a caref...

Combinatorics Enumerative Combinatorics 05

Analytic Combinatorics — Symbolic Combinatorics

This booklet develops in nearly 200 pages the basics of combinatorial enumeration through an approach that revolves around generating functions. The major objects of interest here are words, trees, graphs, and permutations, which surface recurrently in all areas of discrete mathematics. The text presents the core of the theory with chapters on unlabelled enumeration and ordinary generating func...