A collection of counting techniques

• enumeration Techniques

• http://degwer.hatenablog.com/entry/20171220 DEGwer

• PDF

• 2 Summarize the state of the art - DP speeds up all searches - Speeding up the process by equating states - Conditions that can be summarized - Same destination state - The coefficients on the transitions are the same. - All of the summarized states either meet or do not meet the conditions of the problem statement. - ARC059F - codefestival_2016_final_F - AOJ2439

• 3 Change search order

  • 3.1 Sort in descending order of size
    • AOJ2333
  • 3.2 The permutation is insertion DP
    • When determining permutations in DP, it is necessary to keep track of "what has already been used".
      • If it is small, you can use bit DP.
    • Instead of doing it in order, do it in ascending/descending order of what you put in.
  • 3.3 Section sorted by endpoint

• 4 Paraphrasing conditions - Many operations but few products.

• 5 Return from greedy

  • Find the number of columns that may be created by an operation" is difficult when the same column is created by multiple operation columns
  • It would be easier to count if there was a way to uniquely determine the operation sequence from the product.

• 6 Techniques for Case Segregation

  • Case by parameter
    • Be careful to DISJOINT
  • AGC013D

• 7 Decomposition into linear sums

  • It is similar to what I call change order of operations.
    • Decompose bitwise operations into digits

• 8 subgroup techniques - Operation is reversible and the whole area → subgroup - Lagrange's theorem

  • Predicting the number of invariants
    • If it's even, it's likely to have even-odd invariants.

• 9 Use of recursive definitions

  • Works well with DP Recursive definition → DP.
  • ARC037D

• 10 About [Digit DP

  • AOJ0570

• 11 Acceleration Techniques

  • 11.1 Using [cumulative sum
  • 11.2 Use of Data Structures - Fennic tree
  • 11.3 Array usage
  • 11.4 Fast Fourier Transform
    • NTT
  • 11.5 fast zeta transformation
    • Can be used for operations that satisfy the coupling and exchange laws
  • 11.6 Convolution of And and Add - Algorithm similar to [Fast Fourier Transform
  • 11.7 Simple branch trimming

• 12 Techniques with Matrices 26

  • 12.1 power of two
    • AGC013E
    • Power of the companion matrix
    • Power of polynomial
  • 12.2 Matrix Expression Techniques - matrix tree theorem - Number of trees in the whole area - LGV Official - Number of non-intersecting paths
    • https://www.ioi-jp.org/camp/2017/2017-sp_camp-kumabe2.pdf

• 13 Ignore small probabilities. 29

• 14 Binomial coefficient technique 30

  • 14.1 Collection of frequently used formulas - binomial coefficient formula
  • 14.2 Return to [Number of routes
    • Attributing binomial coefficients to the number of paths makes it easy to transform the equation
  • 14.3 45 degree rotation - divide into X and Y
  • 14.4 Catalan number

• 15 principle of inclusion 33

  • 15.1 When symmetry is used
    • AGC005D
  • 15.2 Using DP
  • 15.3 irreducible inclusion
    • ARC064F

• 16 Identifying "unsolvable problems

This page is auto-translated from /nishio/数え上げテクニック集 using DeepL. If you looks something interesting but the auto-translated English is not good enough to understand it, feel free to let me know at @nishio_en. I'm very happy to spread my thought to non-Japanese readers.