Combinatorics of Balanced Carries
نویسندگان
چکیده
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 characters, and the 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.
منابع مشابه
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 ...
متن کاملSome series of block designs with nested rows and columns
Some new series of block designs with nested rows and columns have been constructed with some examples mainly from a combinatorial point of view. Two of the series are balanced (ternary), the others are partially balanced, based on rectangular and triangular association schemes.
متن کاملOn the construction of complete and partial nearest neighbour balanced designs
In this paper, methods for constructing two dimensional nearest neighbour balanced (NNB) designs are considered. The methods given by Afsarinejad and Seeger (1988) are extended to give a new family of nearest neighbour balanced designs. Both nearest neighbour balanced designs with and without borders are constructed. A method of construction of a class of partial nearest neighbour balanced (PNN...
متن کاملTernary codes through ternary designs
It is known that under certain conditions the incidence matrix of a balanced incomplete block design (v, b, r, k, λ) gives a binary code of length b and size 2(v + 1). Here we investigate the conditions where a balanced ternary design gives a similar ternary code.
متن کاملBalanced ternary designs with holes and numbers of common triples
There exists a balanced ternary design with block size 3 and index 2 on 2v P2 + 4 and 2v P2 + 1 elements with a hole of size v, for all positive integers v and P2, such that v ~ 2P2 + 1. As an application of this result, we determine the numbers of common triples in two simple balanced ternary designs with block size 3 and index 2, for P2 = 3 and 4.
متن کامل