Some messy mathematical conclusions.

1. Dislocation Sort:

Dislocation sort sequence: 1,2,9,44,265 ...

Give a 1~n arrangement, now upset it, so that each of the number and its markings are not the same, ask for all the number of disrupted programs

Consider the total number of 1~n, which is

So, when the number on a single is the same as its label (whether the other bits are the same or not), then the number of upset scenarios is

Can see the wrong to take this one out, the remaining number of arbitrary arrangement of the number of programs, total n bits, the overall number is

The discovery will lose more than two people are the same number of cases, to add back

Discovery is a principle of repulsion, just keep pushing.

Formula is

2. Number of Cattleya

Cattleya series: 1,2,5,14 ...

Matrix recursion long like this

The formation of the matrix is like this

1

1 1

1 2 2

1 3 5 5

1 4 9) 14 14

1 5 14 28 42 42

......

A rare sequence of recursion than a general term.

SOURCE formula:

Recursive formula:

General formula:

The concrete derivation is quite complicated

Some topics, such as the number of elements out of the stack, polygon partition triangles and so on

Recently appeared in the NOI2018 bubble sort

Some messy mathematical conclusions.