Enumerative Combinatorics: class notes Darij Grinberg May 4, 2021 (unfinished!) Status: Chapters 1 and 2 finished; Chapter 3 outlined. Contents 1. Introduction7 1.1. Domino tilings . .8 1.1.1. The problem . .8 1.1.2. The odd-by-odd case and the sum rule . 12 1.1.3. The symmetry and the bijection rule . 14 1.1.4. The m = 1 case . 18 1.1.5. The m = 2 case and Fibonacci numbers . 19 1.1.6. Kasteleyn’s formula (teaser) . 26 1.1.7. Axisymmetric domino tilings . 28 1.1.8. Tiling rectangles with k-bricks . 31 1.2. Sums of powers . 38 1.2.1. The sum 1 + 2 + ... + n ....................... 38 1.2.2. What is a sum, actually? . 42 1.2.3. Rules for sums . 48 1.2.4. While at that, what is a finite product? . 53 1.2.5. The sums 1k + 2k + ... + nk .................... 54 1.3. Factorials and binomial coefficients . 58 1.3.1. Factorials . 58 1.3.2. Definition of binomial coefficients . 59 1.3.3. Fundamental properties of the binomial coefficients . 61 1.3.4. Binomial coefficients count subsets . 69 1.3.5. Integrality and some arithmetic properties . 74 1.3.6. The binomial formula . 78 1.3.7. Other properties of binomial coefficients . 86 1.4. Counting subsets . 93 1.4.1. All subsets . 93 1 Enumerative Combinatorics: class notes page 2 1.4.2. Lacunar subsets: the basics . 94 1.4.3. Intermezzo: SageMath . 97 1.4.4. Counting lacunar subsets . 102 1.4.5. Counting k-element lacunar subsets . 107 1.4.6. Counting subsets with a odd and b even elements . 111 1.4.7. The addition formula for Fibonacci numbers . 114 1.4.8. More subset counting . 119 1.4.9. Counting subsets containing a given subset . 126 1.5. Counting tuples and maps . 128 1.5.1. Tuples . 128 1.5.2. Counting maps . 132 1.5.3. Applications . 133 1.6. Interchange of summations . 136 1.6.1. The finite Fubini principle . 136 1.6.2. The Fubini principle with a predicate . 146 1.6.3. A cautionary tale about infinite sums . 156 1.7. Counting permutations: an introduction . 159 1.7.1. Permutations and derangements . 159 1.7.2. Only the size counts . 161 1.7.3. Intermezzo: OEIS . 166 1.7.4. The one-line notation . 168 1.7.5. Short-legged permutations . 172 1.7.6. Long-legged permutations . 176 2. Binomial coefficients 178 2.1. The alternating sum of a row of Pascal’s triangle . 178 2.1.1. Telescoping sums . 178 2.1.2. A war between the odd and the even . 184 2.2. The trinomial revision formula . 187 2.2.1. An algebraic proof . 187 2.2.2. A double counting proof . 191 2.3. The hockey-stick identity revisited . 198 2.4. Counting maps . 202 2.4.1. All maps . 202 2.4.2. Injective maps . 203 2.4.3. The pigeonhole principles . 210 2.4.4. Permutations . 212 2.4.5. Surjective maps . 213 2.5. 1m + 2m + ... + nm .............................. 224 2.6. The Vandermonde convolution . 226 2.6.1. The Vandermonde convolution theorem . 226 2.6.2. The polynomial identity trick . 232 2.6.3. Salvaging the proofs of Theorem 2.6.1 . 236 2.6.4. More consequences of the polynomial identity trick . 238 Enumerative Combinatorics: class notes page 3 2.6.5. Mutating the Chu–Vandermonde identity . 241 2.7. Counting subsets again . 248 2.8. Another use of polynomials . 253 2.9. The Principle of Inclusion and Exclusion . 257 2.9.1. The principles . 257 2.9.2. The cancellation lemma . 264 2.9.3. The proofs . 267 2.9.4. Application: Surjections . 273 2.9.5. Application: Derangements . 276 2.9.6. Application: Euler’s totient function . 282 2.9.7. Other cancellation-type lemmas . 290 2.10. Compositions and weak compositions . 290 2.10.1. Compositions . 291 2.10.2. Binary compositions . 295 2.10.3. Weak compositions . 296 2.10.4. Other composition-like counting problems . 298 2.11. Multisubsets . 299 2.11.1. Definitions . 299 2.11.2. Counting multisubsets of given size . 301 2.11.3. An application to lacunar subsets . 302 2.12. Multinomial coefficients . 307 2.12.1. Definition and formulas . 307 2.12.2. Counting maps that take values a given number of times . 309 2.12.3. Counting anagrams . 312 2.12.4. More formulas . 315 3. The twelvefold way 317 3.1. What is the twelvefold way? . 318 3.2. L ! L ..................................... 323 3.3. Equivalence relations . 324 3.3.1. Relations . 324 3.3.2. Equivalence relations . 326 3.3.3. Equivalence classes . 328 3.3.4. Defining unlabelled boxes and balls . 330 3.4. U ! L .................................... 333 3.5. L ! U .................................... 337 3.6. U ! U and integer partitions . 341 3.7. Integer partitions (an introduction) . 345 3.8. Odds and ends . 348 4. Permutations 349 4.1. Introduction . 349 4.2. Definitions . 350 4.3. Transpositions and cycles . 350 Enumerative Combinatorics: class notes page 4 4.4. Inversions and lengths . 353 4.5. Descents . 359 4.6. Signs . 360 5. Lattice paths (brief introduction) 363 6. Generating functions (introduction) 367 7. Solutions and references to the exercises 369 7.1. Reference to solution to Exercise 1.3.1 . 369 7.2. Solution to Exercise 1.3.2 . 369 7.3. Reference to solution to Exercise 1.3.3 . 369 7.4. Solution to Exercise 1.3.4 . 370 7.5. Reference to solution to Exercise 1.3.5 . 371 7.6. Solution to Exercise 1.3.6 . 371 7.7. Solution to Exercise 1.4.1 . 372 7.8. Solution to Exercise 1.4.2 . 374 7.9. Reference to solution to Exercise 1.4.6 . 375 7.10. Reference to solution to Exercise 1.4.7 . 375 7.11. Reference to solution to Exercise 1.4.8 . 375 7.12. Reference to solution to Exercise 1.5.1 . 375 7.13. Reference to solution to Exercise 1.5.2 . 375 7.14. Reference to solution to Exercise 2.1.1 . 375 7.15. Reference to solution to Exercise 2.2.1 . 376 7.16. Solution to Exercise 2.2.2 . 376 7.17. Reference to solution to Exercise 2.2.3 . 378 7.18. Solution to Exercise 2.4.1 . 378 7.19. Solution to Exercise 2.4.2 . 380 7.20. Solution to Exercise 2.4.3 . 382 7.21. Solution to Exercise 2.4.4 . 383 7.22. Solution to Exercise 2.4.5 . 389 7.23. Solution to Exercise 2.5.1 . ..