Course review of combined mathematics

Specified counting problems are solved by establishing a Bijection Between the set to be counted and some easy-to-count set. This kind of proofs are usually called (non-rigorously)Combinatorial proofs.

 

The numberK-CompositionsNIs equal to the number of solutions to inPositiveIntegers.

Count the number of solutions to inNonnegativeIntegers we call such a solutionWeakK-CompositionOfN.

 

Formally, A MultisetMOn a setSIs a function. For any element, the integer is the number of repetitionsXInM, CalledMultiplicityOfX. The sum of multiplicities is calledCardinalityOfMAnd is denoted as |M|.

We have already evaluated the number. If, letZI=M(XI), Then is the number of solutions to in nonnegative integers, which is the number of weakN-CompositionsK, Which we have seen is.

 

We can think of it as thatNLabeled bils are assignedMLabeled bins, and is the number of assignments such thatI-Th Bin hasAIBils in it.

 

Course review of combined mathematics