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组合学

Combinatorial Gardens

• Mahonian distribution - the number of inversions in a permutation of a multiset...

Combinatorics - Gian-Carlo Rota

• He worked on the theory of incidence algebras (which generalize the 19th-century theory of Möbius inversion) and popularized their study among combinatorialists, set the umbral calculus on a rigorous foundation, unified the theory of Sheffer sequences and polynomial sequences of binomial type, and worked on fundamental problems in probability theory.
• His series of ten papers on the "Foundations of Combinatorics" in the 1960s is credited with making it a respectable branch of modern mathematics.
• Crapo, Rota. On The Foundations of Combinatorial Theory: Combinatorial Geometries.

Twelvefold way

• Twelvefold way compares distinguishable and indistinguishable. Is that like definite and indefinite? How does distinguishable relate to Spencer Brown's Law of Forms?
• Twelvefold way - what if we consider other sets or structures rather than 1,...,n. Does injection become less trivial?
• Threesome (injection, isomorphism, surjection) relates to the combinatorial Twelvefold way. How does isomorphism relate with "any function"?
• Kenneth P. Bogart. "Combinatorics Through Guided Discovery". Twentyfold Way.
• Stanley. Enumerative Combinatorics. Volume I.
• Abhyankar. Enumerative combinatorics of Young tableaux.
• Involution is square root of permutation. Compare with spin as square root of geometry.

Combinatorial principles

• Rule of sum
• Rule of product
• Inclusion-exclusion principle
• Rule of division
• Bijective proof
• Double counting
• Pigeonhole principle
• Method of distinguished element
• Generating function
• Recurrence relation