Height of records in partitions of a set

TL;DR

This paper investigates the properties of strong and weak records in set partitions, providing exact formulas for their counts and heights, advancing combinatorial understanding of partition structures.

Contribution

It introduces new exact formulas for counting strong and weak records in set partitions, extending previous combinatorial results.

Findings

Exact formulas for the number of strong records with height one

Total sum of record heights over set partitions

Number of partitions with a specified maximal height of strong records

Abstract

We study the restricted growth function associated with set partitions, and obtain exact formulas for the number of strong records with height one, the total of record heights over set of partitions, and the number of partitions with a given maximal height of strong records. We also extend some of these results to weak records.

Equations

$$s_{n,k}=-(n-1)s_{n-1,k}+s_{n-1,k-1},n,k\in \mathbb{N},k\le n,$$

$$S_{n,k}=k{S}_{n-1,k}+S_{n-1,k-1},n,k\in \mathbb{N},k\le n,$$

$$\min A_1<\min A_2<\cdots <\min A_k.$$

$$\sum _{j=0}^{k-1}(-1{)}^{r-j}(\genfrac{}{}{0ex}{}{k-1-j}{r-j})|s_{k-1,j}|S_{n-k+1+j,k}.$$

$$\frac{1}{2}{S}_{n+1,k}+\frac{1}{2}{S}_{n,k}-S_{n-1,k}-S_{n-1,k-1}-\frac{1}{2}{S}_{n-1,k-2}.$$

$$\displaystyle\frac{1}{2}B_{n+1}+\frac{1}{2}B_{n}-\frac{5}{2}B_{n-1}=\left(\frac{n+1}{2\xi_{n}}+\frac{1}{2}+O\Bigl{(}\frac{\log n}{n}\Bigr{)}\right)B_{n},$$

$$\displaystyle\xi_{n}=\log n-\log\log n+\frac{\log\log n}{\log n}+O\bigg{(}\frac{\log^{2}\log n}{\log^{2}n}\bigg{)}.$$

$$(k-1)S_{n,k}+\left(\genfrac{}{}{0ex}{}{k}{3}\right)S_{n-1,k}.$$

$$\displaystyle\frac{1}{6}B_{n+2}-\frac{2}{3}B_{n}-\frac{1}{6}B_{n-1}=\bigg{(}\frac{(n+2)(n+1)}{6\xi^{2}}-\frac{2}{3}+O\Bigl{(}\frac{\log n}{n}\Bigr{)}\bigg{)}B_{n}.$$

$$\text{mbox}Card({\mathcal{P}}_{n,k}^h)=\sum _{j=1}^{k-h}{s}_{k-h,j}{S}_{n-(k-h)+j,k}.$$

$$\sum _{i=0}^{k-1}\sum _{a=0}^{(n+i-k)/2}(-1{)}^{m+a-i}(\genfrac{}{}{0ex}{}{n+i-a-k}{a}\left)\right(\genfrac{}{}{0ex}{}{k+a-1-i}{m-i})S_{n+1+i-2a-k,k}|s_{k-1,i}|$$

$$+\sum _{i=0}^{k-1}\sum _{a=0}^{(n+i-1-k)/2}\sum _{j=0}^{n+i-1-2a-k}(-1{)}^{m+a-i}(\genfrac{}{}{0ex}{}{j-1+a}{a}\left)\right(\genfrac{}{}{0ex}{}{k+a-i}{m-i})S_{j,k}|s_{k-1,i}|.$$

$$\frac{1}{2}{S}_{n+2,k}+\frac{1}{2}{S}_{n+1,k}+n{S}_{n-1,k}-S_{n,k-1}-\frac{1}{2}{S}_{n,k-2}-S_{n-1,k}-\sum _{j=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{j}\right)S_{n-1-j,k-1}.$$

$$\frac{1}{2}{B}_{n+1}+\frac{1}{2}{B}_n+(n-1)B_{n-2}-\frac{3}{2}{B}_{n-1}-B_{n-2}-\sum _{j=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n}{j+1}\right)B_j$$

$$\displaystyle\qquad\qquad=\bigg{(}\frac{n+1}{2\xi_{n}}+\frac{1}{2}+O\Bigl{(}\frac{\log n}{n}\Bigr{)}\bigg{)}B_{n}.$$

$$(k-1)S_{n,k}+\left(\genfrac{}{}{0ex}{}{k}{3}\right)S_{n-1,k}+\left(\genfrac{}{}{0ex}{}{k+1}{3}\right)S_{n-2,k}.$$

$$\frac{1}{6}{B}_{n+2}+\frac{1}{6}{B}_{n+1}-\frac{7}{6}{B}_n-\frac{1}{3}{B}_{n-1}+\frac{1}{3}{B}_{n-2}$$

$$\displaystyle\qquad=\bigg{(}\frac{(n+2)(n+1)}{6\xi_{n}^{2}}+\frac{(n+1)}{6\xi_{n}}-\frac{7}{6}+O\Big{(}\frac{\log n}{n}\Big{)}\bigg{)}B_{n}.$$

$$A(x)=\sum _{n\ge 0}{a}_n{x}^n,x\in \mathbb{C},$$

$$\tilde{A}(x):=\sum _{n\ge 0}{a}_n\frac{x^n}{n!}.$$

$$P_k(x)=P_k(x,q_1,\dots ,q_{k-1})=\sum _{n\ge k}{P}_{n,k}{x}^n.$$

$$P_k(x|a_s{a}_{s-1}\cdots a_1)=\sum _{n\ge k}{P}_{n,k}(q_1,\dots ,q_{k-1}|a_s{a}_{s-1}\cdots a_1)x^n,$$

$$P_{n,k}(q_1,\dots ,q_{k-1}|a_s{a}_{s-1}\cdots a_1)$$

$$P_k(x|a)=x{P}_k(x).$$

$$P_k(x)=(k-1)x{P}_k(x)+P_k(x|k).$$

$$P_k(x|k)$$

$$(1-(k-1)x)P_k(x)=x{P}_k(x)+x{q}_1(1-(k-2)x)P_{k-1}(x)$$

$$+x^2(q_2+\cdots +q_{k-1})P_{k-1}(x).$$

$$P_k(x)$$

$$P_k(x;q,q,\dots ,q)=\frac{x^k{q}^{k-1}}{{\prod }_{j=1}^k(1-jx)}=q^{k-1}\sum _{n\ge 0}{S}_{n,k}{x}^n,$$

$$T_k(x;q):=P_k(x;q,1,\dots ,1).$$

$$\displaystyle(1-kx)T_{k}(x;q)=x\big{(}q+(k-2)x(1-q)\big{)}T_{k-1}(x;q)$$

Full text

*MSC2010: * Primary 05A18; Secondary 05A15, 05A16

Keywords: restricted words; partitions of a set; records.

1 Introduction and statement of results

We study records, also known in the literature as left to right maxima. For a sequence $\pi=\pi_{1}\cdots\pi_{n}$ of members of a totally ordered set, $\pi_{i}$ is a strong record if $\pi_{i}>\max_{1\leq j\leq i-1}\pi_{j}$. We call it a weak record if $\pi_{i}\geq\max_{1\leq j\leq i-1}\pi_{j}.$ The interest in records is partially motivated by real-world applications, such as extreme weather studies, tests of randomness, determination of minimal failure, and stresses of electronic components, to name a few. The number of strong records was first studied by Rényi [17] for permutations (see also [9, 11]) where he proved, among other results, that the number of permutations of $[n]$ with $r$ strong records is equal to the number of such permutations with $r$ $k$-cycles, the latter being given by the unsigned Stirling number of the first kind.

More recently, Myers and Wilf [16] extended the study of records to multiset permutations and words. They derived the generating function for the number of permutations of a fixed multiset which contains exactly $r$ (strong or weak) records. They also gave the average number of (strong and weak) records among all permutations of the multiset. Their work has been extended to different totally ordered discrete sets and a variety of statistics of records. For example, Kortchemski in [12] studied the asymptotic behavior of the number of permutations having $r$ records and the permutations for which the sum of the positions of their records is $r$. Knopfmacher, Mansour and Wagner [10] studied the sum of the positions of records in set partitions. Cakić, Mansour and Smith [2] investigated further another statistic of records in set partitions. Asakly [1] obtained an explicit formula and asymptotic estimate for the total number of sum of weighted records over set partitions of $[n]$ in terms of Bell numbers.

In this paper we study the height of strong and weak records in set partitions. Most of our results are exact and obtained by analyzing underlying generating functions. The asymptotic behavior of the quantities of interest can be subsequently deduced from the exact formulas.

We begin with notations. A partition of a set $A$ is a collection of non-empty, mutually disjoint subsets, called blocks, whose union is the set $A$. A partition $\Pi$ with $k$ blocks is called a $k$-partition and denoted by $\Pi=A_{1}\mid A_{2}\mid\ldots\mid A_{k}$. Let $\mathcal{P}_{n,k}$ be the set of $k$-partitions of $[n]=\{1,2,\ldots,n\}$ and define $\mathcal{P}_{n}=\cup_{k=0}^{n}\mathcal{P}_{n,k}$ to be the collection of all partitions of $[n]$. It is well-known that $\mbox{Card}(\mathcal{P}_{n,k})=S_{n,k}$ and $\mbox{Card}(\mathcal{P}_{n})=B_{n},$ where $S_{n,k}$ is a Stirling number of the second kind and $B_{n}$ is the $n$-th Bell number. The Stirling numbers of the first kind $s_{n,k}$ and the second kind $S_{n,k}$ can be introduced algebraically in several different ways, including formulas (10) and (11) given below. Alternatively, one can define the sequence of Stirling numbers of the first kind as the solution to the recursion

[TABLE]

with the initial conditions $s_{0,0}=1$ and $s_{0,n}=0,$ and the sequence of Stirling numbers of the second kind as the solution to the recursion

[TABLE]

with the initial conditions $S_{0,0}=1$ and $S_{0,n}=0.$ The sequence of Bell numbers $(B_{n})_{n\geq 0}$ can be then defined through the formula $B_{n}=\sum_{k=0}^{n}S_{n,k},$ or, for instance, recursively via the formula $B_{n+1}=\sum_{k=0}^{n}\binom{n}{k}B_{k}$ with the initial condition $B_{0}=1.$ For more on the Stirling and Bell numbers see, for instance, [5, 14, 18] and references therein.

A $k$-partition $\Pi$ is said to be in the standard form if the blocks $A_{i}$ are labeled in such a way that

[TABLE]

Note that if a $k$-partition of $[n],$ call it $\Pi,$ is in the standard form, then it can be represented equivalently by the canonical sequential form $\Pi=\pi_{1}\pi_{2}\ldots\pi_{n}$ where $\pi_{i}\in[n]$ and $i\in A_{\pi_{i}}$ for all $i$ (see, for instance, [14, 18]). In words, $\pi_{i}$ is the label of the partition block that contains $i.$ We remark that the canonical form is sometimes referred as a restricted growth function (see, for instance, [14]).

Notices that (1) guarantees that $1\in A_{1}.$ For example, the canonical representation of the partition $\{1,4,7\}\mid\{2,3,6,9\}\mid\{5,8\}\in\mathcal{P}_{9,3}$ is $122132132.$ Throughout this paper we consider the canonical form as a representation of partitions. Note that a word $\pi\in[k]^{n}$ is a canonical representation of a $k$-partition of $[n]$ if and only if the following holds:

Each element of $[k]$ appears at least once in $\pi.$ 2. 2.

For all $1\leq i<j\leq k,$ the first occurrence of $i$ precedes the first occurrence of $j$.

To define records for a $k$-partition $\Pi$ of $[n]$, we represent $\Pi$ by its corresponding restricted growth function $\pi=\pi_{1}\cdots\pi_{n}$. Then, we define the records of $\Pi$ as records of their restricted growth function $\pi$. For a record, $\pi_{i},$ we call $i$ the position of the record $\pi_{i}.$ Similarly, we define the height of this record as $\pi_{i}-\pi_{i-1}$ whenever $i>1$ and (within the framework of strong records) [math] if $i=1$.

Our first result is

Theorem 1.1**.**

**

Let $n\geq k\geq 1$ and $r\geq 0$. The number of $k$-partitions of $[n]$ with exactly $r$ records of height one is

[TABLE]

The total number of strong records of height one in all partitions in $\mathcal{P}_{n,k}$ is given by

[TABLE]

The total number of strong records of height one in all partitions in $\mathcal{P}_{n}$ is given by

[TABLE]

where $\xi_{n}$ is the unique positive root of the equation $\xi e^{\xi}=n+1.$

We remark that (see, for instance, Example VIII.6 in [7])

[TABLE]

Similarly to Theorem 1.1, next theorem provides some insight on strong records with the exception that we drop the assumption of height one.

Theorem 1.2**.**

**

The sum of the heights of the strong records over all the partitions in $\mathcal{P}_{n,k}$ is given by

[TABLE]

The sum of the heights of the strong records over all the partitions in $\mathcal{P}_{n}$ is given by

[TABLE]

Let $\mathcal{P}^{h}_{n,k}$ be the set of partitions in $\mathcal{P}_{n,k}$ whose maximum height is $h.$ Then

[TABLE]

Next theorem extends our previous results to weak records.

Theorem 1.3**.**

**

Let $n\geq k\geq 1$ and $r\geq 0$. The number of $k$-partitions of $[n]$ with exactly $r$ weak records of height one is given by

[TABLE]

The total number of weak records of height one in all the partitions in $\mathcal{P}_{n,k}$ is given by

[TABLE]

The total number of weak records of height one in all the partitions in $\mathcal{P}_{n}$ is given by

[TABLE]

The sum of the heights of the weak records over all the partitions in $\mathcal{P}_{n,k}$ is given by

[TABLE]

The sum of the heights of the weak records over all the partitions in $\mathcal{P}_{n}$ is given by

[TABLE]

The rest of the paper is organized as follows. Theorems 1.1 and 1.2 are proved in Section 2, and the proof of Theorem 1.3 is given in Section 3. The proofs are analytical and rely on the analysis of generating functions.

2 Strong records. Proof of Theorems 1.1 and 1.2

In this section we are concerned with the height of strong records. The section is divided into four subsection. Section 2.1 is of a preliminary function, its goal is to introduce certain generating function associated with strong records. The proof of Theorem 1.1 is given in Section 2.2. The proof of claims (i) and (ii) of Theorem 1.2 is contained in Section 2.3, and the proof of part of the theorem is included in Section 2.4.

2.1 Preliminaries

We first introduce a few notations. Throughout this paper, for any given ordinary generating function

[TABLE]

we use $\widetilde{A}$ to denote the corresponding exponential generating function. That is,

[TABLE]

We define $P_{n,k}=P_{n,k}(q_{1},q_{2},\ldots,q_{k-1})$ to be the generating function of the number of partitions in $\mathcal{P}_{n,k}$ with respect to the height of the strong records, where $q_{i}$ counts the number of records of the height $i$. For $k\geq 0$ define

[TABLE]

We remark that $P_{0}(x)=1$ and $P_{1}(x)=\frac{x}{1-x}$. Thus, unless explicitly said otherwise, for the rest of this paper we assume that $k\geq 2$.

In order to study the generating function $P_{k}(x)$ we define

[TABLE]

where

[TABLE]

is the generating function for the number of $k$-partitions of the form $\pi=\pi^{\prime}a_{s}a_{s-1}\cdots a_{1}$ in $\mathcal{P}_{n,k}$. Observe that the last symbol in the canonical representation of a $k$-partition cannot be a strong record unless it equals to $k.$ Hence, for $a\in[k-1]$ we have

[TABLE]

Using the fact that $P_{k}(x)=\sum_{a=1}^{k}P_{k}(x|a),$ we obtain the identity

[TABLE]

Furthermore,

[TABLE]

where the last equality follows from (5), and the empty sum $\sum_{i=2}^{k-1}q_{i}$ is considered to be zero when $k=2.$ It follows then from (6) that

[TABLE]

Equivalently,

[TABLE]

We remark that if $q_{1}=\cdots=q_{k-1}=q$, then (7) becomes

[TABLE]

where the second equality is due to (11). This is expected because $\mbox{Card}(\mathcal{P}_{n,k})=S_{n,k}$ and any set $k$-partition of $[n]$ has exactly $k-1$ strong records.

2.2 Strong records of height one. Proof of Theorem 1.1

The goal os this section is to prove Theorem 1.1. In order to analyze the number of strong records of height one, we let $q_{1}=q$ and $q_{i}=1$ for all $i\geq 2$ in (7). Denote

[TABLE]

Then (7) yields the recursion

[TABLE]

with $T_{0}(x;q)=1$ and $T_{1}(x;q)=\frac{x}{1-x}$. Thus,

[TABLE]

Recall (see, for instance, [5, 14]) that Stirling numbers satisfy the identities

[TABLE]

(note that the latter identity is equivalent to the former one with $x=-1/y$), and

[TABLE]

Therefore, (9) can be expressed as

[TABLE]

Changing the order of summation from $\sum_{n\geq k}\sum_{j=0}^{k-1}\sum_{i=0}^{k-1-j}...$ to

[TABLE]

and inspecting the coefficient of $q^{r}x^{n}$ in the resulting formula for $T_{k}(x;q)$ finishes the proof of Theorem 1.1-(i).

Next, we prove Theorem 1.1-(ii) and  (iii). Let $\widetilde{T}$ be the exponential generating function associated with $T,$ as defined in (4). Then, by (8), we get

[TABLE]

with $\widetilde{T}_{0}(x;q)=1$ and $\widetilde{T}_{1}(x;q)=e^{x}-1$. Define

[TABLE]

Multiplying by $y^{k}$ and summing up over $k\geq 2$ gives

[TABLE]

Note that $\widetilde{T}(x,y;1)=e^{y(e^{x}-1)}.$ By taking the derivative in the both sides of (13) and evaluating the result at $q=1,$ we obtain the equation

[TABLE]

with the initial condition

[TABLE]

Therefore,

[TABLE]

Inspecting the coefficient of $x^{n}y^{k}/n!$ in $\frac{\partial}{\partial q}\widetilde{T}(x,y;q)\mid_{q=1}$ and the coefficient of $x^{n}/n!$ in $\frac{\partial}{\partial q}\widetilde{T}(x,1;q)\mid_{q=1}$ finishes the proof of Theorems 1.1-(ii) and (iii).

We remark that in order to derive the asymptotic estimate in Theorem 1.1-(iii), we use the fact that (see, for instance, [3]) the identity

[TABLE]

holds uniformly for $h=O(\log n)$, where $\xi_{n}$ is the unique positive root of $\xi e^{\xi}=n+1$. ∎

2.3 Total height of strong records. Proof of Theorem 1.2-(i),(ii)

The goal of this section is to prove part (i) and part (ii) of Theorem 1.2. In order to analyze the sum of heights of all strong records over $P_{k,n}$, we let $q_{i}=q^{i}$ in (7). Define $T_{k}(x;q):=P_{k}(x;q,q^{2},\ldots,q^{k-1})$ to be the generating function for the number of partitions of $\mathcal{P}_{n,k}$ accounting for the total heights of all strong records. In this notation, (7) becomes

[TABLE]

Take the derivative with respect to $q$ on the both sides of (15) and evaluate the result at $q=1$. Then

[TABLE]

Recall (11). An argument similar to the one given in the previous section finishes the proof of Theorem 1.2-(i).

Before we proceed with the proof of part (ii) of Theorem 1.2, we recall another identity for $S_{n,k}$. We refer the reader to [14] for a review of this type of identities for Stirling’s numbers. Recall $\frac{(e^{x}-1)^{k}}{k!}=\sum_{n\geq k}S_{n,k}\frac{x^{n}}{n!}.$ Then

[TABLE]

Define \widetilde{T}_{k}(x):=\frac{\partial}{\partial q}T_{k}(x;q)\big{|}_{q=1}. Then, (15) implies that

[TABLE]

In particular,

[TABLE]

The inspection of the power series expansion of the above expression for $\frac{\partial}{\partial x}\sum_{k\geq 0}\widetilde{T}_{k}(x)$ reveals that the coefficient of $\frac{x^{n}}{n!}$ is given by

[TABLE]

This completes the proof of Theorem 1.2-(ii). Note that the asymptotic estimate in (3) is a straightforward consequence of the exact formula and (14). ∎

2.4 Maximum height of strong records. Proof of Theorem 1.2-(iii)

We now turn to the proof of Theorem 1.2-(iii). Let $\mathcal{P}^{h}_{n,k}$ be the set of partitions in $\mathcal{P}_{n,k}$ whose maximum height is $h$. To obtain the generating function of $\mbox{Card}(\mathcal{P}^{h}_{n,k}),$ we let $q_{i}=1$ for $i\in[h]$ and $q_{i}=0$ for $i\geq h+1$ in (7), thus setting

[TABLE]

Hence,

[TABLE]

It follows then from (10) and (11) that the number of partitions in $\mathcal{P}^{h}_{n,k}$ is given by

[TABLE]

This completes the proof of Theorem 1.2-(iii). ∎

3 Weak records. Proof of Theorem 1.3

This section is devoted to to prove Theorem 1.3. The proof of claims (i)–(iii) of the theorem is given in Section 3.2. The proof of claims (iv) and (v) is included in Section 3.3. Section 3.1 is of a preliminary function, its goal is to compute a certain generating function associated with weak records.

3.1 Generating functions

The goal of this section is to obtain Lemma 3.1, a closed form expression for certain generating functions. Let $Q_{n,k}=Q_{n,k}(q_{1},q_{2},\ldots,q_{k-1})$ be the multivariate generating function for the number of partitions in $\mathcal{P}_{n,k}$ accounting for the heights of weak records. Here $q_{i}$ counts the number of the weak records of height $i$. Define

[TABLE]

It is easy to see that

[TABLE]

From now we will assume that $k\geq 2$. In order to study $Q_{k}(x)$, we introduce auxiliary generating functions

[TABLE]

where $Q_{n,k}(q_{1},\ldots,q_{k-1}|a_{s}a_{s-1}\cdots a_{1})x^{n}$ is the generating function for the number of partitions $\pi=\pi^{\prime}a_{s}a_{s-1}\cdots a_{1}$ in $\mathcal{P}_{n,k}$ accounting for the heights of the weak records.

Let $\pi=\pi^{\prime}a$ be a partition with $k$ blocks and $1\leq a\leq k-1$. Since the partition is assumed to be in the standard form and $a<k,$ it cannot be a record. Thus

[TABLE]

Therefore,

[TABLE]

In order to compute $Q_{k}(x|k)$, we consider $k$-partitions in the form $\pi=\pi^{\prime}k$ and two separate scenarios: when $\pi^{\prime}$ is a $k$-partition and when it is a $(k-1)$-partition. We have:

[TABLE]

With the help of (17), this formula simplifies to

[TABLE]

It then follows from (18) that

[TABLE]

Finally, by induction on $k$ we get

Lemma 3.1**.**

For $k\geq 1$,

[TABLE]

We remark that the particular case of (19) with $q_{1}=\cdots=q_{k-1}=q,$ namely

[TABLE]

has been calculated in [10].

3.2 Weak records of height one. Proof of Theorem 1.3-(i)–(iii)

In order to study the weak records of height one, we will utilize Lemma 3.1 with $q_{1}=q$ and $q_{i}=1$ for all $i\geq 2$. It follows from (19) that

[TABLE]

Next, using (10) and (11), we obtain:

[TABLE]

Thus, the coefficient of $q^{r}$ in $Q_{k}(x;q,1,\ldots,1)$ is

[TABLE]

Finally, inspecting the coefficient of $x^{n}$ we obtain Theorem 1.3-(i).

Next, we prove Theorem 1.3-(ii) and (iii). To this end, let $q_{1}=q$ and $q_{i}=1$ for all $i\geq 2$ in (3.1). Then

[TABLE]

This can be written recursively as

[TABLE]

Let $\widetilde{Q}$ be the exponential generating function associated with $Q,$ as defined by (4). Then (20) gives the following integral equation:

[TABLE]

Differentiating twice with respect to $x,$ we arrive to the following second order ODE:

[TABLE]

with initial conditions

[TABLE]

Define $\widetilde{Q}(x,y;q)=\sum_{k\geq 0}\widetilde{Q}_{k}(x;q)y^{k}$. Multiplying both sides of (21) by $y^{k}$ and summing over $k\geq 2$, we get

[TABLE]

Let $\widetilde{T}(x,y)=\frac{\partial}{\partial q}\widetilde{Q}(x,y;q)\mid_{q=1}$. Since $\widetilde{Q}(x,y;1)=e^{y(e^{x}-1)}$, (22) gives

[TABLE]

Finally, using the initial condition $\frac{\partial}{\partial x}\widetilde{T}(0,y)=0$ we arrive to

[TABLE]

Inspecting the coefficient of $x^{n}y^{k}/n!$ and the coefficient of $x^{n}/n!$ in $\widetilde{T}(x,y)$ and $\widetilde{T}(x,1)$ finishes the proof of Theorem 1.3-(ii) and (iii). ∎

3.3 Total height of weak records. Proof of Theorem 1.3-(iv),(v)

Let $q_{i}=q^{i}$ for all $i\geq 1$ in (19). Therefore,

[TABLE]

Taking the derivative with respect to $q$ and evaluating it at $q=1,$ we obtain that

[TABLE]

which by virtue of (11) implies the claim in part (iv) of Theorem 1.3.

We turn now to the proof of part (v) of the theorem. We start with an auxiliary algebraic identity. Recall (see, for instance, [5] or [14]) that

[TABLE]

Using this fact we can calculate

[TABLE]

In order to calculate the total of all heights of all weak records over $\mathcal{P}_{n}$, we take advantage of the corresponding exponential generating function of $T_{n}(x)$ as defined by (4). Summing up $\widetilde{T}_{k}(x)y^{k}$ over all $k,$ taking the second derivative of the summation with respect to $x,$ and leveraging the previous combinatorial identity, we get

[TABLE]

Finally, extracting the coefficient of $x^{n}/n!$ in $\frac{\partial^{2}}{\partial x^{2}}\sum_{k\geq 0}\widetilde{T}_{k}(x)$ gives Theorem 1.3-(v). ∎