let mapping
[permutation] The following are equivalent
- bijection
- permutation
- order permutation. Quantity . Usually denoted as
[combination] The following are equivalent
- Select subset with from
- Select subsets with
- Select a permutation with and another permutation gives the same partition if
Define the [quotient] of permutations with the same partition as a subset of that satisfies the above conditions, i.e., the inverse image of the partitioning possibility
<==>
Cardinality calculation
Denoted as
All combinations <==> All select subsets with in
is the number of repeatable selections of from
from can be calculated by induction or observed directly, we can get
[binom-expansion]
vs Newton binomial
[multi-combination] Similarly, the following are equivalent
- multi-combination. select with
- partition with and
-
Select permutation, and quotient
Total number , which is the number of selecting times from with replacement
Proof
Selecting times with replacement from , number <==> number of mappings ,
Any selection can be permuted to with
Restoring all the order is selecting positions from , which is multinomial coefficient . This gives
what is ?
Example Quantity 10, number of groups 4.
star & bar model
โ โ โ | โ โ | | โ โ โ โ โ
Selecting positions from positions as bar, dividing โ into groups. Number
[dimension-of-symmetric-tensor] The dimension of symmetric tensor space is also , basis
The repetition number will be used for example to calculate normalization
[conjugate-class-of-permutation-is-cycle] conjugate-class of <==> cycle := permutation with
The decomposition of permutation
Tensor space's decomposition, irreducible represenation