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

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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

Curtis Greene, Albert Nijenhuis, and Herbert S. Wilf, A probabilistic proof of a formula for the number of Young tableaux of a given shape, Adv. in Math. 31 (1979), no. 1, 104–109. MR 521470, DOI 10.1016/0001-8708(79)90023-9 Michel Goemans, Math 18.310A (2015) is a rather eclectic discrete mathematics class; it includes notes on counting and on generating functions. 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 G. de B. Robinson, On the Representations of the Symmetric Group, Amer. J. Math. 60 (1938), no. 3, 745–760. MR 1507943, DOI 10.2307/2371609

Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol.576, New York Acad. Sci., New York, 1989, pp. 500–535. MR 1110850, DOI 10.1111/j.1749-6632.1989.tb16434.x Kevin Purbhoo's 630 notes ("Algebraic Enumeration") introduce various more advanced topics such as species, symmetric functions, representations of symmetric groups and cycle index functions. Mohr M (2020b) Manfred Mohr’s vimeo video channel. https://vimeo.com/manfredmohr. Accessed 24 Sept 2020

Janés C (2014) Juan Eduardo Cirlot. Cuando la palabra y la letra llaman a su forma. Tintas. Quaderni di letterature iberiche e iberoamericane Special issue pp 389–400 Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: The art of combinatorial proof, The Dolciani Mathematical Expositions, vol.27, Mathematical Association of America, Washington, DC, 2003. MR 1997773

Doing it the right way

Drew Armstrong, Discrete mathematics, 2019. An introductory undergraduate class that includes the basics of enumerative combinatorics (up until Prüfer codes for trees) and of graph theory, probably quite appropriate as an appetizer before most of the other texts here. Armstrong writes in an intriguing conversational way that exposes not just proofs but also motivations and thought processes (as well as tidbits of tangential context). 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 Vriezen S (2017) Diagrams, games and time (Towards the analysisi of open form scores). In: Pareyon G, Pina-Romero S, Agustín-Aquino OA, Lluis-Puebla E (eds) The musical-mathematical mind. Patterns and transformations. Springer, Cham Markus Fulmek, VO Combinatorics, 2018. Graduate topics course covering species, asymptotics and posets. Richard P. Stanley, Differential posets, J. Amer. Math. Soc. 1 (1988), no. 4, 919–961. MR 941434, DOI 10.1090/S0894-0347-1988-0941434-9

David M. Burton, The history of mathematics: An introduction, 2nd ed., W. C. Brown Publishers, Dubuque, IA, 1991. MR 1223776

Counting permutations

This is the number of k k k-permutations from a set of size n n n, but its not too easy on the eyes! Fortunately, we can make it more presentable, since the value of the expression remains unchanged when it’s multiplied and divided by the same quantity: 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. 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: 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. Fritz D (2011) Images and sound created and synchronized by algorithm Vladimir Bonačić’s computer-generated interactive audiovisual object GF. E(16,4), 1969–1974. In: Proceedings of the International Conference Pierre Schaeffer: MediArt, Rijeka, 2011. Museum of Modern and Contemporary Art, Rijeka, pp 134–143

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995), no. 1, 166–194. MR 1317387, DOI 10.1006/aima.1995.1020

There might be other scenarios

Dominique Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol.31, Reidel, Dordrecht-Boston, Mass., 1977, pp. 27–49 (French). With a comment by Richard P. Stanley. MR 519777 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 Bernhard Riemann, Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge, Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], Suppl. 1, BSB B. G. Teubner Verlagsgesellschaft, Leipzig; Springer-Verlag, Berlin, 1990 (German). Based on the edition by Heinrich Weber and Richard Dedekind; Edited and with a preface by Raghavan Narasimhan. MR 1066697, DOI 10.1007/978-3-663-10149-9