Indexing schemes of binary sequences - MathOverflow

Indexing schemes of binary sequences

2010-07-13T22:04:33Z

I am looking for "low-complexity" indexing methods to enumerate binary sequences of a given length and a given weight.

Formally, let $T_k^n = \{x_1^n \in \{0,1\}^n: \sum_{i=1}^n x_i = k\}$. How to construct a bijective mapping $f: T_k^n \to \{1, 2, \ldots, \binom{n}{k}\}$ such that computing each $f(x_1^n)$ needs small number of operations?

For example, one could do lexicographical ordering, that is, e.g., $0110 < 1010$. Then this gives the following scheme:

$f(x_1^n) = \sum_{k=1}^n x_k \binom{n-k}{w_k}$

where $w_k=\sum_{i=k}^n x_i$. Computing $n$ binomial coefficients can be quite demanding. Any other ideas? Or is it impossible to avoid?

Answer by Steven Stadnicki for Indexing schemes of binary sequences

2010-07-13T23:08:14Z

You want Volume 4, Fascicle 3 of Knuth's The Art of Computer Programming, chapter 7.2.1.3: "Generating All Combinations" - I won't include links because everyone has a favorite online bookseller, but AFAIK all the major ones have it in stock. Highly, highly recommended!

Answer by mathy for Indexing schemes of binary sequences

2010-07-14T08:57:03Z

You'll probably want to use the combinatorial number system or combinadic. La Wik has a useful overview and a number of references including Knuth's The Art of Computer Programming