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arXiv:math/9205211math.HO

Two notes on notation

Donald E. Knuth

Abstract

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.

arXiv comments
Abstract added by Greg Kuperberg
Journal reference
Amer. Math. Monthly 99 (1992), no. 5, 403--422
DOI
Subjects
math.HOHistory and Overview

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Linked by 12 identifiable commenters in 13 comments across 10 threads, from May 2013 to Jun 2021. Found in 4 HN submissions from Jun 2019 to Jun 2026. Open submission discussions.

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Discussed in comments10 threads

  1. Two notes on notation (1992) 2 comments

    • Original link is broken. Use this one. https://arxiv.org/abs/math/9205211

      threatofrainView on HNParent context

    • Note that the abstract at https://arxiv.org/abs/math/9205211 is not by Knuth (it says "Abstract added by Greg Kuperberg" in small print). Anyway, the "two notes" of the title are about: 1. Iverson bracket notation: writing say "[x ≥ 5]" to mean "1 if x ≥ 5, and 0 otherwise"…

      svatView on HNParent context

  2. Floor, Ceiling, Bracket 1 comment

    • Knuth [0] discusses in more detail some advantages of indicator notation, in particular that, when you eliminate the limits on summations, it makes it much easier to do algebra on summations. [0] https://arxiv.org/abs/math/9205211

      tlbView on HNParent context

  3. A History of APL in Fifty Functions (2016) 1 comment

    • I don't know if Knuth popularized it, but it's somewhat common in combinatorics. Similar idea to using indicator functions when doing change of variables in calculus. https://arxiv.org/abs/math/9205211

      dubyaView on HNParent context

  4. Geometry, Algebra, and Intuition (2017) 1 comment

    • This seems almost exactly backwards to me. “Geometric” proofs (i.e. based on spatial reasoning) are fast and fluid, relying on an imagined spatial configuration which can be seen all at once. “Algebraic” proofs (i.e. based on symbol manipulation) are serialized and…

      jacobolusView on HNParent context

  5. Boiling lead and black art: a history of mathematical typography 1 comment

    • My point was that the notation should be optimized for the people using it, not for the technology used to typeset it. If the notation is suboptimal for the people then it can definitely be changed, but that's a different reason. Note that some of these notational improvements…

      svatView on HNParent context

  6. Ask HN: What is your favorite CS paper? 1 comment
  7. A Quirky Function 2 comments

    • Here's Knuth's opinion on 0^0: http://arxiv.org/pdf/math/9205211v1.pdf (on page 6) He makes a few pretty convincing points about why it's useful to define it equal to 1, while admitting that in some sense, it's not quite as well defined as say 0+0.

      joefkelleyView on HNParent context

    • This is a great example. Wikipedia has the same one with isolines demonstrating this point in the 0^0 subsection of the exponentiation article, here: https://en.m.wikipedia.org/wiki/Exponentiation#Continuous_ex... With the caveat: "However, under certain conditions, such as when…

      dahartView on HNParent context

  8. Defining zero factorial 1 comment

    • Defining 0^0 = 1 is just as important in analysis as it is in combinatorics, even though analysis texts aren’t always clear that they’re making use of this fact. • If p(x) = ∑[n=0..∞] a_n x^n is a power series, then p(0) = a_0 is its constant term, but it makes no sense to write…

      anderskaseorgView on HNParent context

  9. 0^0 2 comments

    • A word from Knuth on the matter (warning: PDF): http://arxiv.org/pdf/math/9205211v1.pdf See page 6.

      yaks_hairbrushView on HNParent context

    • Why link to the PDF? Better to just link to the abstract: http://arxiv.org/abs/math/9205211 Edit: To be clear, this is good practice when linking to arXiv in general. From the abstract, one can easily click through to the PDF; not so the reverse. And the abstract allows one to…

      SniffnoyView on HNParent context

  10. PEP 435 Accepted – Adding an Enum type to the Python standard library 1 comment

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