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

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Arthur Benjamin and Jennifer Quinn, Proofs that Really Count. This looks rather introductory (focusing on proofs of identities involving binomial coefficients or Fibonacci numbers using bijections). ( Suggested by JSchlather.)

J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric groups, Canad. J. Math. 6 (1954), 316–324. MR 62127, DOI 10.4153/cjm-1954-030-1 Miklós Bóna, Combinatorics of permutations, 2nd ed., with a foreword by Richard Stanley, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012. MR 2919720, DOI 10.1201/b12210

There might be other scenarios

William Fulton, Young tableaux, CUP 1997. Part I and Appendix A are purely combinatorial, and (in my opinion) the most readable source on a lot of semistandard tableau theory.

Peter J. Cameron, Notes on Counting goes deeper into enumeration than his above-mentioned Notes on Combinatorics. A draft version is available online. 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: Winter M (2019) A few more thoughts about Leibniz: the prediction of harmonic distance in harmonic space. MusMat Braz J Music Math 3:79–92 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

Bruce Sagan, Mathematician

Bernt Lindström, On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973), 85–90. MR 335313, DOI 10.1112/blms/5.1.85

Gian-Carlo Rota, On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368 (1964). MR 174487, DOI 10.1007/BF00531932 Bill M (1996) The mathematical approach in contemporary art. In: Stiles K, Selz P (eds) Theories and documents of contemporary art. A Sosurcebook of Artists’ Writings, University of California Press, Berkeley, pp 74–77 Chen Chuan-Chong and Koh Khee-Meng, Principles and Techniques in Combinatorics is another text that approaches the subject through olympiad problems. Carolina Benedetti, Joshua Hallam, and John Machacek, Combinatorial Hopf algebras of simplicial complexes, SIAM J. Discrete Math. 30 (2016), no. 3, 1737–1757. MR 3543152, DOI 10.1137/15M1038281 This chapter is motivated by a question I asked myself: “How can combinatorial structures be used in a work of art?” Immediately, other questions arose: Are there artists that work or think combinatorially? If so, what works have they produced in this way? What are the similarities and differences between artworks produced using combinatorics? Combinatorics is a very transversal branch of mathematics. It is connected to logic, and it intervenes in the building of languages in general as natural language, musical language, and poetry. Combinatorics has been present in artistic practice for millenia, especially in music and poetry; for instance, in the shaped poems of Simmias of Rhodes in Ancient Greece. However, we are interested in artistic practices that are driven by the use of combinatorial ideas, structures, and methodologies, artists whose work is conceptualized or merely inspired by combinatorics. This often happens in connection with an interest in structure and language, and the phenomenon was significant in the twentieth century, as a by-product of the revolution that took place not only in art but in many other areas of knowledge. In this chapter, we present a survey of artists that think combinatorially, work combinatorially, and construct combinatorial artworks. The selection covers music, literature, visual arts including digital art, and an example of early physical interactive art, dance, theatre, and cinema. It is a non-exhaustive list of artists, selected to show differences and similarities in their ways of approaching art when using combinatorics. KeywordsJoseph P. S. Kung, Gian-Carlo Rota, Catherine H. Yan, Combinatorics: The Rota Way, CUP 2009 is a nonstandard textbook, originating from Rota's MIT classes and reflecting his interests (many of which have never hit the combinatorial mainstream). Errata and a sample chapter.