Combinatorics: The Art of Counting (Graduate Studies in Mathematics)

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Keefer C, Guldemond J (2013) Oskar Fischinger 1900–1967. In: Experiments in cinematic abstraction. EYE Filmmuseum, Amsterdam

Andrés Aranda, Manuel Bodirsky, Combinatorics. These are notes for what would probably be a late-undergrad topics class (in Germany, a third year bachelor course). They cover flows/cuts, probabilistic method, Ramsey theory, generating functions. There is a more elementary prequel ( Diskrete Strukturen) in German. Both are works in progress and welcome corrections. Zweig J (1997) Ars combinatoria: mystical systems, procedural art, and the computer. Art J 56(3):20–29Peter Doubilet, Gian-Carlo Rota, and Richard Stanley, On the foundations of combinatorial theory. VI. The idea of generating function, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 267–318. MR 0403987 Curtis Greene, An extension of Schensted’s theorem, Advances in Math. 14 (1974), 254–265. MR 354395, DOI 10.1016/0001-8708(74)90031-0 Donald E. Knuth, The Art of Computer Programming was started in 1962 as an attempt at a comprehensive textbook for programming. Four volumes are out by now, thus giving computer science its own Song of Ice and Fire to wait on. Combinatorics (enumerative and algorithmic and occasionally even algebraic) is everywhere dense in them (at least in Volumes 1, 3 and 4A), and Knuth's propensity to wildly curious digressions (one gets the impression that he even digresses from digressions) makes these books an incredibly addictive nerd-read. Volume 3 is the one most relevant to algebraic combinatorics, with its §5.1 devoted to permutations and tableaux. Few authors have dug as deep as Knuth into the history of the subject -- witness a draft of §7.2.1.7. newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\) C. Krattenthaler, Advanced determinant calculus: a complement, Linear Algebra Appl. 411 (2005), 68–166. MR 2178686, DOI 10.1016/j.laa.2005.06.042

Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, Representation theory of the symmetric groups, CUP 2010. A slow-paced introduction to symmetric functions and representations of symmetric groups, taking a rather nonstandard approach. (Particularly recommended due to the inclusion of partition algebras, which are currently underexposed in textbooks.) Joshua Hallam, Applications of quotient posets, Discrete Math. 340 (2017), no. 4, 800–810. MR 3603561, DOI 10.1016/j.disc.2016.11.019 Bruce E. Sagan, Congruences via abelian groups, J. Number Theory 20 (1985), no. 2, 210–237. MR 790783, DOI 10.1016/0022-314X(85)90041-1 Petter Brändén, Unimodality, log-concavity, real-rootedness and beyond, Handbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2015, pp. 437–483. MR 3409348

Stephan Wagner, Combinatorics is yet another set of notes. Generating functions appear to be the red thread here. Seems terse, though. n × ( n − 1 ) × ( n − 2 ) × ⋯ × ( n − ( k − 1 ) ) n \times (n-1) \times (n-2) \times \cdots \times (n- (k-1)) Titu Andreescu and Zuming Feng, A Path to Combinatorics for Undergraduates: Counting Strategies. Another introduction to enumerative combinatorics. This one takes a problem-solving approach, illustrating principles on olympiad-style problems. ( Suggested by Marko Amnell.) D. Laksov, A. Lascoux, P. Pragacz, and A. Thorup, The LLPT notes. This is a fairly readable introduction to the combinatorial part of Schubert calculus and the work of Lascoux and others. (Don't worry about the missing sections; nothing depends on them.)

Fritz D (2008) Vladimir Bonačić: computer-generated works made within Zagreb’s new tendencies network (1961–1973). Leonardo 41(2):175–183 I don't know these books/notes well enough to tell which of them are better suited for a first course (although I don't have any reasons to suspect any of them to be unsuitable), but it cannot hurt to try each of them and go as far as you can before meeting serious resistance. (And once you meet serious resistance, either keep going or try the next one.) Half of these are freely available (and so are the other half, if you search in the darker places).A. P. Hillman and R. M. Grassl, Reverse plane partitions and tableau hook numbers, J. Combinatorial Theory Ser. A 21 (1976), no. 2, 216–221. MR 414387, DOI 10.1016/0097-3165(76)90065-0 Miklos Bona, Introduction to Enumerative Combinatorics, 2007. This is another Bona book, and explicitly directed at undergraduates, though it does percolate into some advanced topics as well (unimodality, magic square enumeration). A 2nd edition has come out in 2016 under the extended title Introduction to Enumerative and Analytic Combinatorics (adding, as the name change suggests, a chapter on analytic combinatorics). Errata. Knowing the difference between permutations and combinations is, without exaggeration, essential knowledge for most engineering disciplines. Here are some of the ways in which the concepts explored in this blog are applied in different kinds of content on the Educative platform: Richard P. Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 336–356. MR 409206, DOI 10.1016/0097-3165(76)90028-5 P. A. MacMahon, The Indices of Permutations and the Derivation Therefrom of Functions of a Single Variable Associated with the Permutations of any Assemblage of Objects, Amer. J. Math. 35 (1913), no. 3, 281–322. MR 1506186, DOI 10.2307/2370312