Sets of iterated Partitions and the Bell iterated Exponential Integers

TL;DR

This paper introduces a new set-theoretic framework for nested partitions that generalizes Bell and Stirling numbers, revealing their higher-order structures and combinatorial properties.

Contribution

It defines higher order Stirling numbers through nested partitions and explores their combinatorial and set-theoretic properties, extending classical partition concepts.

Findings

Cardinality of nested partition sets equals higher order Bell numbers.

Defined higher order Stirling numbers via set-theoretic partitions.

Explored the combinatorics of hyper partition sets.

Abstract

It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain partitioning process that gives rise to a sequence of sets of "nested" partitions. We prove that at stage m, the cardinality of the resulting set will equal the m-th order Bell number. This set-theoretic interpretation enables us to make a natural definition of higher order Stirling numbers and to study the combinatorics of these entities. The cardinality of the elements of the constructed "hyper partition" sets are explored.

Tables

Table 1. Table 1: Higher order Bell numbers B n ( m ) superscript subscript 𝐵 𝑛 𝑚 B_{n}^{(m)} when 1 ≤ n ≤ 8 1 𝑛 8 1\leq n\leq 8 and 1 ≤ m ≤ 5 1 𝑚 5 1\leq m\leq 5

$m\backslash n$	1	2	3	4	5	6	7	8
1	1	2	5	15	52	203	877	4140
2	1	3	12	60	358	2471	19302	167894
3	1	4	22	154	1304	12915	146115	1855570
4	1	5	35	315	3455	44590	660665	11035095
5	1	6	51	561	7556	120196	2201856	45592666

Table 2. Table 2: Some examples of higher order Stirling and Bell numbers

$m$	$S^{\left(m\right)}(5,1)$	$S^{\left(m\right)}(5,2)$	$S^{\left(m\right)}(5,3)$	$S^{\left(m\right)}(5,4)$	$S^{\left(m\right)}(5,5)$	$B_5^{\left(m\right)}$
5	3455	3325	725	50	1	7556
20	1115320	233050	11900	200	1	1360471
50	45533300	3706375	74750	500	1	49314926

Equations

$$E_m(x)=\sum _{n=0}^{\infty }{B}_n^{(m)}\frac{x^n}{n!}(m\ge 1),$$

$$|{\wp }_n^{(m)}|=B_n^{(m)}.$$

$$|{\wp }_n^{(m+1)}|=\sum _{k=1}^n|{\wp }_k^{(m)}|S(n,k),|{\wp }_k^{(0)}|=1.$$

$$P^{(m+1)}(x)=\sum _{k=1}^{\infty }|{\wp }_k^{(m)}|\sum _{n=1}^{\infty }S(n,k)\cdot \frac{x^n}{n!}.$$

$$P^{(m+1)}(x)=\sum _{k=1}^{\infty }|{\wp }_k^{(m)}|\cdot \frac{(e^x-1{)}^k}{k!}=P^{(m)}(e^x-1).$$

$$B_n^{(m+1)}=\sum _{k=1}^n{B}_k^{(m)}S(n,k),B_k^{(0)}=1,$$

$$B_n^{(m)}=\sum _{s=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{s}\right)B_s^{(m)}{B}_{n-s}^{(m-1)},B_0^{(m)}=1,$$

$$B_n=\sum _{s=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{s}\right)B_s.$$

$$S^{(m)}(n,k)={\zeta }_n^{(k,m)},$$

$$\frac{(E_{m-1}(t)-1{)}^k}{k!}=\sum _n{\zeta }_n^{(k,m)}\cdot \frac{t^n}{n!}.$$

$$F_k^{(m)}(x)\equiv \sum _{n=k}^{\infty }{S}^{(m)}(n,k)\frac{x^n}{n!}.$$

$$S^{(m+1)}(n,k)=\sum _{i=k}^n{S}^{(m)}(i,k)S(n,i).$$

$$F_k^{(m+1)}(x)=\sum _{n,i}{S}^{(m)}(i,k)S(n,i)\frac{x^n}{n!}=\sum _i{S}^{(m)}(i,k)\frac{(e^x-1{)}^i}{i!}=F_k^{(m)}(e^x-1),$$

$$F_k^{(m+1)}(x)=F_k^{(m)}(e^x-1).$$

$$F_k^{(m)}(x)=\frac{(E_{m-1}(x)-1{)}^k}{k!},(E_0(x)=e^x),$$

$$P^{(m)}(x)=\sum _{n=1}^{\infty }|{\wp }_n^{(m)}|\frac{x^n}{n!}=\sum _{k=1}^{\infty }{F}_k^{(m)}(x)=E_m(x)-1.$$

$$S^{(m)}(n,k)=\sum _{i=k}^n{S}^{(m-r)}(i,k)S^{(r)}(n,i),$$

$$B_n^{(m)}=\sum _{i=1}^n{B}_i^{(m-r)}{S}^{(r)}(n,i),$$

$$S^{(m)}(n,k)=\sum _{s=k-1}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{s}\right)B_{n-s}^{(m-1)}{S}^{(m)}(s,k-1)$$

$$S^{(m)}(n,k)=\frac{1}{k!}\sum _{i_1+\cdots +i_k=n}\left(\genfrac{}{}{0ex}{}{n}{i_1,\dots ,i_k}\right)B_{i_1}^{(m-1)}\cdots B_{i_k}^{(m-1)},$$

$$B_n^{(m)}=\sum _{k=1}^n\frac{1}{k!}\sum _{i_1+\cdots +i_k=n}\left(\genfrac{}{}{0ex}{}{n}{i_1,\dots ,i_k}\right)B_{i_1}^{(m-1)}\cdots B_{i_k}^{(m-1)}.$$

$${\left\{\sum _{i_1+\cdots +i_k=n}\left(\genfrac{}{}{0ex}{}{n}{i_1,\dots ,i_k}\right)B_{i_1}^{(m-1)}\cdots B_{i_k}^{(m-1)}\right\}}_{n=0}^{\infty }.$$

$$F_k^{(m)}(x)=\frac{1}{k!}{f}_{m-1}^k(x)\text{mbox}and{f}_m(x)=\sum _{k=1}^{\infty }\frac{f_{m-1}^k(x)}{k!}=\exp (f_{m-1}(x))-1.$$

$$B_n^{(m)}=c_{n-1}{m}^{n-1}+c_{n-2}{m}^{n-2}+\cdots +c_0,$$

$$\underset{m\to \infty }{\lim }\frac{B_n^{(m)}}{B_n^{(m-1)}}=1,$$

$$A_n^{(m)}=\frac{1}{B_n^{(m)}}\sum _{k=1}^{n}k{S}^{(m)}(n,k)\underset{m\to \infty }{⟶}1.$$

$$S^{(m)}(n,k)=o(B_n^{(m)}),k\ge 2(m\to \infty ),$$

Full text

1 A partitioning process.

Consider the 3-set $S=\{a,b,c\}$. The partition set $\wp_{3}^{(1)}$ of $S$, where the elements of $S$ are put into boxes, contains the five partitions shown in the second column of Figure 1.

Now we proceed, putting boxes into boxes. This means that we create a second order partition set to each first order partition in $\wp_{3}^{(1)}$. The union of all the second order partition sets is denoted by $\wp_{3}^{(2)}$, and appears in the third column of Figure 1.

Definition**.**

$\wp_{n}^{(1)}$ is the set of all partitions of a given $n$-set. For $m>1$, $\wp_{n}^{(m)}$, called the $m$-th order partition set, is the union of the complete collection of sets, each being the partition set of an element in $\wp_{n}^{(m-1)}$.

We observe that the number of partitions in $\wp_{3}^{(2)}$ is $|\wp_{3}^{(2)}|=12$, which also, and not by coincidence, turns out to be the second order Bell number $B_{3}^{(2)}$.

2 Connecting higher order Bell numbers and hyper partitions.

The $m$-th order Bell numbers $B_{n}^{(m)}\quad(n=0,1,\ldots)$, studied by E. T. Bell in [1], are given by the exponential generating functions

[TABLE]

where $E_{1}(x)=\exp(\exp(x)-1)$ and $E_{m+1}(x)=\exp(E_{m}(x)-1)$.

In Table 1 $B_{n}^{(m)}$ is computed for a few values of $m$ and $n$.

Theorem 1**.**

The number of $m$-th order partitions of an $n$-set is $B_{n}^{(m)}$ ($m,n\geq 1$), i.e.

[TABLE]

Proof.

The proof makes use of generating functions. Recall that there are altogether $S(n,k)$ (Stirling number of the second kind) distinct $k$-partitions of the given $n$-set, i.e. there are $S(n,k)$ elements in $\wp_{n}^{(1)}$ which are $k$-sets. Each of these $k$-sets gives rise to $|\wp_{k}^{(m)}|$ distinct partitions of order $m+1$ of the $n$-set we started with, i.e. elements in $\wp_{n}^{(m+1)}$. Furthermore, different elements in $\wp_{n}^{(1)}$ of course give different elements in $\wp_{n}^{(m+1)}$, since they are already different at the ”ground level”. This means that we have the recurrence formula

[TABLE]

Now, let $P^{(m)}(x)=\sum_{n=1}^{\infty}|\wp_{n}^{(m)}|\cdot x^{n}/n!$ denote the exponential generating function of $\{|\wp_{n}^{(m)}|\}_{n=1}^{\infty}$. Multiplication with $x^{n}/n!$, summation over $n$ and changing the order of summation in (2), leads to

[TABLE]

It is well known that $\sum_{n=1}^{\infty}S(n,k)\cdot x^{n}/n!=(e^{x}-1)^{k}/k!$ is the exponential generating function of the Stirling numbers of the second kind. (A proof of this fact is included in Example 4.) From (3) we therefore get the recurrence formula

[TABLE]

Now, since $P^{(0)}(x)=e^{x}-1$, a straightforward induction argument yields $P^{(m)}(x)=E_{m}(x)-1$, which proves (1) as well as the relation

[TABLE]

which now follows from (2). ∎

Example 1**.**

To demonstrate the power of hyper partition thinking, we give a combinatorial proof of the relation

[TABLE]

that appeared in [1, p. 545, (2.11)].

Each element (i.e. a hyper partition) in $\wp_{n}^{(m)}$ consists of nested sets where the sets at the ground level are subsets of the basic $n$-set $S$. For a partition $p\in\wp_{n}^{(m)}$ we call elements of $S$ related if they ”reside” in the same outer set (box) in $p$. Now, fix an arbitrary $a\in S$. We count the number of partitions in $\wp_{n}^{(m)}$ according to which elements $a$ is related: Let $A$ be a basic $(n-s)$-set, including $a$, of related elements. Now for each of these $\binom{n-1}{n-s-1}=\binom{n-1}{s}$ sets there are $B_{n-s}^{(m-1)}$ inner structures. For the complementary $s$-set $C=S\setminus A$, the partitioning process will generate $B_{s}^{(m)}$ partitions of order $m$. By combining the possibilities, (6) follows.

Observe that if we in the same manner as above fix two (or more) elements in $S$, new formulas emerge.

By putting $B_{n}^{(0)}=1$, we note that (6) is a generalized version of the well known formula (see [3, p. 210])

[TABLE]

3 Higher order Stirling numbers.

Having established the relationship between partitions of order $m$ and higher order Bell numbers, defining higher order Stirling numbers seems like a natural thing to do.

Definition**.**

The $m$-th order Stirling number $S^{(m)}(n,k)$ (of the second kind) is the number of $k$-sets in $\wp_{n}^{(m)}$.

In [2] E. T. Bell gave an analytical definition of what he called generalized Stirling numbers $\zeta_{n}^{(k,m)}$ by means of generating functions. In Theorem 2 we prove that $S^{(m)}(n,k)=\zeta_{n}^{(k,m)}$.

We note that the higher order Stirling numbers $S^{(m)}(n,k)$ are the entries of the matrix $\textbf{S}^{m}$, where $\textbf{S}=(S(n,k))$. This can be seen by induction from the relation (7) in the proof of Theorem 2. Table 2 is computed with the aid of such matrices.

Theorem 2**.**

Let $S^{(m)}(n,k)$ be the $m$-th order Stirling numbers of the second kind. Then we have

[TABLE]

where $\zeta_{n}^{(k,m)}$ are the generalized Stirling numbers of the second kind, defined by E.T. Bell in [2, p. 91] by the generating functions

[TABLE]

Proof.

Let $F_{k}^{(m)}$ denote the generating function of $\{S^{(m)}(n,k)\}_{n=k}^{\infty}$. Then we have

[TABLE]

The idea is to come up with an analogous formula to (2), in order to obtain an analogous formula to (4). We claim that

[TABLE]

This is true because we know that each of the $S(n,i)$ first order partitions will generate $S^{(m)}(i,k)$ $k$-partitions of order $m+1$. Multiplication by $x^{n}/n!$ followed by summation over $n$ in (7) gives

[TABLE]

because $\sum_{n}S(n,i)x^{n}/n!=(e^{x}-1)^{i}/i!$. We have thus deduced the recurrence formula

[TABLE]

Since $F_{k}^{(1)}(x)=(e^{x}-1)^{k}/k!$, induction yields

[TABLE]

which completes the proof. ∎

We notice that Theorem 1 is proved once more since we have $|\wp_{n}^{(m)}|=\sum_{k=1}^{n}S^{(m)}(n,k)$ and

[TABLE]

How the introduction of the higher order Stirling numbers opens up the scope for hyper partition thinking, is illustrated in the next examples.

Example 2**.**

If we, in our construction process of $\wp_{n}^{(m)}$, stop at the $r$-th intermediate stage, i.e. in $\wp_{n}^{(r)}$, making up status so far by grouping the elements according to their cardinality before advancing further on, we get the following generalized version of (7):

[TABLE]

since there are $S^{(r)}(n,i)$, $i$-sets in $\wp_{n}^{(r)}$. This also follows from the matrix representation $\textbf{S}^{m}=\textbf{S}^{m-r}\textbf{S}^{r}$, as well as from (7) by induction.

Summing from $k=1$ to $n$ yields

[TABLE]

which generalizes (5).

Example 3**.**

The formula

[TABLE]

may be proved combinatorially in just the same manner as (6) in Example 1. Note that (8) yields (6) by summation over $k$.

Example 4**.**

Counting the $k$-sets in $\wp_{n}^{(m)}$ by first forming the $k$ ”families” (outer sets) of related elements, we get

[TABLE]

where in this case $B_{k}^{(0)}=1,\;k\geq 1$ and $B_{0}^{(m)}=0$, because a family with $i$ relatives yields $B_{i}^{(m-1)}$ elements in $\wp_{i}^{(m-1)}$, i.e. there are exactly $B_{i}^{(m-1)}$ possible ”inner” structures for an ”$i$-family”.

Summing over $k$ yields

[TABLE]

Note that we have used nothing but the set-theoretic hyper partition interpretation/definition of $B_{n}^{(m)}$ and $S^{(m)}(n,k)$ in establishing (9) and (10). Now, therefore, let $f_{m-1}(t)=f_{m-1}$ be the exponential generating function (e.g.f.) to $\{B_{n}^{(m-1)}\}_{n=0}^{\infty}$. And observe then that $f_{m-1}^{k}$ is the e.g.f. of the sequence

[TABLE]

Now we multiply (9) and (10) with $x^{n}/n!$, sum over $n$ and change the order of summation, to obtain

[TABLE]

And since $f_{0}(x)=e^{x}-1$, we have $f_{m}(x)=E_{m}(x)-1$. So by this reasoning ”from outside in”, we have an alternative proof of Theorem 1 as well as of Theorem 2.

4 An asymptotic consideration

In [1, p. 545] E.T. Bell proved the interesting formula

[TABLE]

where $c_{n-1},\ldots,c_{0}$ are rational numbers, independent of $m$. When $n$ is fixed, this implies that

[TABLE]

enabling us to say something more about the cardinality of the members of $\wp_{n}^{(m)}$.

Remark**.**

Via a slightly different proof of (11) than in [1] one might show that $c_{n-1}=n!/2^{n-1}$, see [4].

When looking at the ”children” of a $p\in\wp_{n}^{(m)}$, i.e. the elements in $\wp_{n}^{(m+1)}$ that $p$ gives rise to, we see that all but one of them have lower cardinality than their ”father” $p$. Following the next generations in the partitioning process, it appears that the great majority of the descendants are 1-element sets (see Table 2). It is therefore easy to conjecture that the average value $A_{n}^{(m)}$ of the cardinality of the sets in $\wp_{n}^{(m)}$ approaches 1 as $m\rightarrow\infty$, i.e.

[TABLE]

Let us see why (13) is true. (12) yields, in conjunction with the observation $S^{(m)}(n,1)=B_{n}^{(m-1)}$, that $S^{(m)}(n,1)\sim B_{n}^{(m)}\quad(m\rightarrow\infty)$. Since $B_{n}^{(m)}=\sum_{k=1}^{n}S^{(m)}(n,k)$, we consequently have

[TABLE]

and (13) follows immediately.

5 A summary comment

The set-theoretic intepretation of the higher order Bell numbers places them in a natural and fundamental context. Together with the higher order Stirling numbers, they take the same central position in the combinatorics of the higher order partition sets as their predecessors $B_{n}$ and $S(n,k)$ have at the ground level.