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June 12, 2015

Combinatorics Course at Cheenta | Problem Discussion

This is a collection of some revision notes. They include topics discussed in first three sessions of Combinatorics Course at Cheenta (Faculty: Ashani Dasgupta).

combinatorics 1(work sheet)

  • Study of symmetry in geometry is greatly facilitated by combinatorial methods
    • There are 6 symmetries of an equilateral triangle (=3! permutations of 3 things)
    • There are 8 symmetries of a square (8 out of 4! permutations of 4 things are used up)
    • All 24 symmetries (including orientations) of a tetrahedron account for 4! permutations of 4 things
  • Cycle notation helps in exploiting permutations
    • Length of a cycle equals it's order
  • Bijection Principle helps to count sets which are otherwise difficult to count.
    • Number of non negative integer solutions of a + b + c + d = n is \displaystyle{ \binom{n+3}{3}}
    • We use balls and bars technique to do this
  • Partitions
    • Conjugate Partitions and Ferrar's diagram
    • Catalan Numbers

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