Discrete Mathematics II — Combinatorics

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• Modeling a counting question ~9 min
• Distinguishable versus identical objects ~9 min
• Counting by cases ~9 min
• Avoiding double counting ~9 min
• Overcounting and dividing out ~9 min
• Double counting as a technique ~9 min
• Counting by bijection ~9 min
• Constructing the pigeonhole boxes ~9 min
• Sanity-checking a count ~8 min
• Assembling a counting argument ~10 min

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• Permutations of a multiset ~9 min
• Circular arrangements ~8 min
• Necklaces and reflection symmetry ~9 min
• Selections with repetition ~9 min
• Stars and bars with bounds ~9 min
• Compositions of an integer ~8 min
• Lattice paths ~8 min
• Catalan numbers ~9 min
• The distribution taxonomy ~9 min
• Choosing an arrangement model ~9 min

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• Pascal's identity revisited ~9 min
• The binomial theorem generalized ~9 min
• Vandermonde's identity ~9 min
• The hockey-stick identity ~8 min
• Absorption and subset identities ~8 min
• Combinatorial proof of binomial identities ~9 min
• Multinomial coefficients revisited ~9 min
• The multinomial theorem ~9 min
• Counting multinomial expansion terms ~8 min
• Symmetric identities from the binomial theorem ~8 min
• Choosing an identity-proving strategy ~9 min

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• The two-set formula revisited ~8 min
• Three or more sets ~9 min
• The general inclusion-exclusion formula ~9 min
• Proving inclusion-exclusion by counting ~8 min
• Counting with forbidden positions ~8 min
• The sieve viewpoint ~7 min
• Derangements ~9 min
• Counting surjections by inclusion-exclusion ~8 min
• Euler's totient by inclusion-exclusion ~8 min
• Choosing when inclusion-exclusion pays off ~8 min

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• Functions between finite sets, counted ~8 min
• Counting injective functions ~8 min
• Counting surjective functions ~9 min
• Stirling numbers of the second kind ~8 min
• The recurrence for Stirling numbers ~9 min
• Set partitions and Stirling numbers ~9 min
• Bell numbers ~8 min
• The Bell number recurrence ~9 min
• Counting equivalence relations ~8 min
• Choosing among function, partition, and relation counts ~8 min

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• What a combinatorial proof requires ~8 min
• Proving Pascal's identity combinatorially ~8 min
• Proving the hockey-stick identity combinatorially ~8 min
• Proving a subset-sum identity combinatorially ~8 min
• Bijective proof as a technique ~8 min
• Building a bijection between two counted sets ~9 min
• Algebraic versus combinatorial proof ~8 min
• When a combinatorial proof is the clearer proof ~8 min
• Constructing your own combinatorial proof ~9 min

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• First-order linear recurrences ~9 min
• Nonhomogeneous recurrences ~8 min
• Particular solutions by undetermined coefficients ~9 min
• The characteristic equation revisited ~9 min
• Repeated characteristic roots ~8 min
• Complex characteristic roots, a first look ~8 min
• Recurrences from counting problems ~9 min
• Verifying a closed form by induction ~8 min
• Recurrence versus closed form ~7 min
• Recursion trees for recurrences ~8 min
• Choosing a recurrence-solving method ~8 min

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• Divide-and-conquer algorithms as recurrences ~8 min
• The recursion-tree method ~9 min
• The iteration/unrolling method ~9 min
• The master theorem ~9 min
• Master theorem case 1: leaves dominate ~8 min
• Master theorem case 2: balanced levels ~8 min
• Master theorem case 3: the root dominates ~8 min
• Binary search and merge sort as recurrences ~8 min
• Comparing recurrence-solving methods ~8 min

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• Ordinary generating functions defined ~8 min
• From sequence to generating function ~8 min
• From generating function to sequence ~8 min
• The algebra of generating functions ~9 min
• The generating function for a geometric sequence ~8 min
• Solving recurrences with generating functions ~9 min
• The Fibonacci generating function ~9 min
• Generating functions for counting problems ~9 min
• The generating function for compositions ~9 min
• Generating functions and the binomial theorem ~8 min
• Choosing a technique: recurrence or generating function ~8 min

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• What an integer partition is ~8 min
• Ferrers diagrams ~7 min
• Conjugate partitions ~8 min
• The partition function ~8 min
• Partitions into distinct parts ~8 min
• Partitions into odd parts ~7 min
• Generating functions for partitions ~9 min
• A partition identity proved bijectively ~9 min
• Restricted partitions ~8 min

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• Counting labeled graphs ~8 min
• Counting simple graphs by degree ~8 min
• Regular graphs ~8 min
• Degree sequences and graphical sequences ~8 min
• Counting the spanning trees of a small graph ~9 min
• Cayley's formula, stated ~8 min
• A bijective flavor of Cayley's formula ~9 min
• The matrix-tree theorem, stated ~9 min
• Counting trees versus counting graphs ~8 min
• Choosing an enumeration approach ~8 min

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• Bipartite graphs ~8 min
• Matchings ~8 min
• Maximum matchings ~8 min
• Hall's marriage theorem ~9 min
• Applications of Hall's theorem ~8 min
• Graph coloring ~8 min
• The chromatic number ~9 min
• Chromatic polynomials ~9 min
• Counting proper colorings ~8 min
• Independent sets and cliques ~8 min
• Extremal counting: how large can a structure be ~9 min