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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 "[$\mkern1mu$P$\mkern1mu$]" 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. [\n] 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.

@article{knuthTwoNotesNotation1992, title = {Two Notes on Notation}, author = {Knuth, Donald E.}, year = {1992}, month = may, volume = {99}, pages = {403--422}, issn = {0002-9890}, doi = {10.2307/2325085}, 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 "[{$\mkern1mu$}P{$\mkern1mu$}]" 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. [\textbackslash n] 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.}, archivePrefix = {arXiv}, eprint = {math/9205211}, eprinttype = {arxiv}, journal = {The American Mathematical Monthly}, keywords = {*imported-from-citeulike-INRMM,~INRMM-MiD:c-7011016,~to-add-doi-URL,iverson-bracket,mathematics,notation,notation-as-a-tool-of-thought}, lccn = {INRMM-MiD:c-7011016}, number = {5} }

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