# 数学代写|组合学代写Combinatorics代考|Ordered partitions and the Fubini numbers

## 数学代写|组合学代写Combinatorics代考|The definition of the Fubini numbers

The number of partitions of an $n$ element set is given by the $n$th Bell number. If the elements in the block are ordered, we get the $L_n$ numbers the horizontal sums of the Lah numbers. If, in turn, the order of the blocks is taken into account, we arrive at another counting sequence: that of the Fubini numbers. In this chapter we study this sequence; it will turn out that it is connected also to some interesting counting problems regarding permutations.
Without taking the order of the blocks into account, the partitions
$$1,5|2,3| 4,6$$
and
$$2,3|1,5| 4,6$$
are identical. In this chapter we do distinguish them and make the following definition.

Definition 6.1.1. Let $F_n$ denote the number of all the partitions of an $n$ set such that we take the order of the blocks in the individual partitions into account. $F_n$ is the $n$th ordered partition number, or $n$th Fubini ${ }^1$ number. The Fubini numbers are also called ordered Bell numbers ${ }^2$.

Going back to our very first example in the book, we saw that four elements have 15 partitions:
$$\begin{gathered} 1|2| 3 \mid 4 \ 1|2| 3,4, \quad 1|3| 2,4, \quad 1|4| 2,3, \quad 2|3| 1,4, \quad 2|4| 1,3, \quad 3|4| 1,2 \ 1|2,3,4, \quad 2| 1,2,4, \quad 3|1,2,4, \quad 4| 1,2,3, \quad 1,2|3,4, \quad 1,3| 2,4, \quad 1,4 \mid 2,3 \ 1,2,3,4 \end{gathered}$$

## 数学代写|组合学代写Combinatorics代考|Two more interpretations of the Fubini numbers

There are other ways we can look at the ordered partitions. We give here some examples.
Surjective functions
Going back to p. 17 , we see there that the number of surjective functions from an $n$-set to a $k$-set is $k !\left{\begin{array}{l}n \ k\end{array}\right}$. Summing over $k$, we infer that the Fubini numbers count all the surjective functions on an $n$-set onto some set (of cardinality necessarily between one and $n$ ).

Another interpretation comes if we consider a competition of $n$ people where draws (or ties) are allowed. The $n$th Fubini number $F_n$ gives the number of the possible outputs in such a competition. This is so because an ordered $k$ partition on $n$ people can be considered as an output of a competition where the $n$ people (better to say, their indices) in the same blocks are classified equal (draws).

This interpretation offers a recursive formula for the Fubini numbers. To enumerate all the outputs of our competition, we can choose $k$ competitors from $n$, and these go to the first position. This can be done in $\left(\begin{array}{l}n \ k\end{array}\right)$ ways. Then the remaining $n-k$ competitors can be considered as competitors of a new competition for the second, third,… position. They can be classified in $F_{n-k}$ ways. Summing over $k$, put these into a formula:
$$F_n=\sum_{k=1}^n\left(\begin{array}{l} n \ k \end{array}\right) F_{n-k}$$

# 组合学代写

## 数学代写|组合学代写Combinatorics代考|The definition of the Fubini numbers

$$1,5|2,3| 4,6$$

$$2,3|1,5| 4,6$$

$$1|2| 3|41| 2|3,4, \quad 1| 3|2,4, \quad 1| 4|2,3, \quad 2| 3|1,4, \quad 2| 4|1,3, \quad 3| 4|1,21| 2,3,4,$$

## 数学代写|组合学代写Combinatorics代考|Two more interpretations of the Fubini numbers

$$F_n=\sum_{k=1}^n(n k) F_{n-k}$$

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