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Donald E. Knuth

Publications and source records attributed to Donald E. Knuth.

29 records · Page 2Linked to original sources
arXiv1992

Johann Faulhaber and sums of powers

Early 17th-century mathematical publications of Johann Faulhaber contain some remarkable theorems, such as the fact that the $r$-fold summation of $1^m,2^m,...,n^m$ is a polynomial in $n(n+r)$ when $m$ is a positive odd number. The present paper explores a computation-based approach by which Faulhaber may well have discovered such results, and solves a 360-year-old riddle that Faulhaber presented to his readers. It also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers. Faulhaber's coefficients can moreover be generalized to factorial powers of noninteger exponents, obtaining asymptotic series for $1^α+2^α+...+n^α$ in powers of $n^{-1}(n+1)^{-1}$.

math.CA↗
arXiv1992

The problem of compatible representatives

The purpose of this note is to attach a name to a natural class of combinatorial problems and to point out that this class includes many important special cases. We also show that a simple problem of placing nonoverlapping labels on a rectangular map is NP-complete.

cs.DS↗
arXiv1992

Convolution polynomials

The polynomials that arise as coefficients when a power series is raised to the power $x$ include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically.

math.CA↗
arXiv1992

Two notes on notation

The author advocates two specific mathematical notations from his popular course and joint textbook, "Concrete Mathematics". The first of these, extending an idea of Iverson, is the notation "[P]" for the function which is 1 when the Boolean condition P is true and 0 otherwise. This notation can encourage and clarify the use of characteristic functions and Kronecker deltas in sums and integrals. The second notation puts Stirling numbers on the same footing as binomial coefficients. Since binomial coefficients are written on two lines in parentheses and read "n choose k", Stirling numbers of the first kind should be written on two lines in brackets and read "n cycle k", while Stirling numbers of the second kind should be written in braces and read "n subset k". (I might say "n partition k".) The written form was first suggested by Imanuel Marx. The virtues of this notation are that Stirling partition numbers frequently appear in combinatorics, and that it more clearly presents functional relations similar to those satisfied by binomial coefficients.

math.HO↗
arXiv1991

Context-free multilanguages

This article is a sketch of ideas that were once intended to appear in the author's famous series, "The Art of Computer Programming". He generalizes the notion of a context-free language from a set to a multiset of words over an alphabet. The idea is to keep track of the number of ways to parse a string. For example, "fruit flies like a banana" can famously be parsed in two ways; analogous examples in the setting of programming languages may yet be important in the future. The treatment is informal but essentially rigorous.

cs.DS↗
arXiv1991

Theory and practice

The author argues to Silicon Valley that the most important and powerful part of computer science is work that is simultaneously theoretical and practical. He particularly considers the intersection of the theory of algorithms and practical software development. He combines examples from the development of the TeX typesetting system with clever jokes, criticisms, and encouragements.

cs.GL↗
arXiv1991

Textbook examples of recursion

We discuss properties of recursive schemas related to McCarthy's ``91 function'' and to Takeuchi's triple recursion. Several theorems are proposed as interesting candidates for machine verification, and some intriguing open questions are raised.

cs.CC↗
arXiv1991

Efficient representation of perm groups

This note presents an elementary version of Sims's algorithm for computing strong generators of a given perm group, together with a proof of correctness and some notes about appropriate low-level data structures. Upper and lower bounds on the running time are also obtained. (Following a suggestion of Vaughan Pratt, we adopt the convention that perm $=$ permutation, perhaps thereby saving millions of syllables in future research.)

math.GR↗
arXiv1990

Nested satisfiability

A special case of the satisfiability problem, in which the clauses have a hierarchical structure, is shown to be solvable in linear time, assuming that the clauses have been represented in a convenient way.

cs.CC↗
arXiv1990

Stable husbands

Suppose $n$ boys and $n$ girls rank each other at random. We show that any particular girl has at least $({1\over 2}-ε) \ln n$ and at most $(1+ε)\ln n$ different husbands in the set of all Gale/Shapley stable matchings defined by these rankings, with probability approaching 1 as $n \to \infty$, if $ε$ is any positive constant. The proof emphasizes general methods that appear to be useful for the analysis of many other combinatorial algorithms.

math.CO↗