note-math

let mapping

[permutation] the following are equivalent

• a bijection from to itself, called a permutation
• an ordering of
• permutation of order . quantity . usually denoted as

[combination] the following are equivalent

• picking a subset from with
• picking subsets with

• picking a permutation with
  another permutation gives the same partition if

  define the [quotient] for permutations giving this same partition, the inverse image of partition

cardinality calculation of the quotient set

denoted as

all combinations <==> picking subset from with for all

is the number of ways to pick times from items with replacement

calculating by induction or direct observation, we get

[binom-expansion]

vs Newton binomial

[multi-combination] similarly, the following are equivalent

• -fold combination. picking from with
• partition with and

• picking a permutation, and quotient

Total number , which is the number of ways to choose times from items with repetition

It can be understood like this. Choosing with repetition times, number <==> number of mappings , which is

The mapping is partitioned into outputting of , satisfying

Partitioning into , outputting of respectively, the number corresponds to multinomial combinations

what is ?

[dimension-of-symmetric-tensor] Also gives the dimension of the symmetric tensor space as , with basis

The repetition count is used for things like calculating normalization

[conjugate-class-of-permutation-is-cycle] conjugate-class of <==> cycle with same partition

cycle := permutation with

The decomposition of permutations

decomposition of tensor spaces, irreducible represenation