Arnold Knopfmacher

University of the Witwatersrand | wits · School of Mathematics

A partition of a positive integer (Formula presented.) is a non-increasing sequence of positive integers whose sum is (Formula presented.). It may be represented by a Ferrers diagram. These diagrams contain corners which are points of degree two. We define corners of types (Formula presented.), (Formula presented.) and (Formula presented.), and als...

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtain...

Consider Motzkin paths which are lattice paths in the plane starting at the origin, running weakly above the x-axis and after n unit steps returning at the point (n,0). The allowed steps are the up and down steps (1,1) and (1,-1) respectively and certain horizontal steps. We consider two types of horizontal steps that have attracted recent attentio...

We consider samples of n geometric random variables with parameter 0<p<1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property t...

We define the push statistic on permutations and multipermutations and use this to obtain various results measuring the degree to which an arbitrary permutation deviates from sorted order. We study the distribution on permutations for the statistic recording the length of the longest push and derive an explicit expression for its first moment and g...

A matching point in compositions and words is an extension to these objects of the well-studied concept of fixed points in permutations. The equivalent of the derangement problem is solved here by providing a formula for the number of compositions of n having no matching points, and showing that the number of words with no matching points tends to...

We study fixed points in integer partitions viewed, respectively, as weakly increasing or weakly decreasing structures. A fixed point is a point with value i in position i. We also study matching points in weakly decreasing partitions. These are defined as positions where the partition and its reverse have the same size parts. From the generating f...

We define an alphabetic point in a restricted growth function as a point in the word which is larger than all preceding points and smaller than all following ones. We find the trivariate generating function for restricted growth functions which tracks the number of alphabetic points as the main statistic. The generating function is then used to fin...

Compositions (ordered partitions) of n are finite sequences of positive integers that sum to n. We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the concept of protected cells and protected columns in the bargraph repres...

Композиции числа $n$ - это такие конечные последовательности положительных целых чисел $(\sigma_i)_{i=1}^k$, что $$ \sigma_1+\sigma_2+\cdots +\sigma_k=n. $$ Композиция $n$ представляется в виде гистограммы площади $n$: высота $i$-го столбца гистограммы равна величине $i$-й части композиции. Мы рассматриваем клеточный периметр гистограммы, который р...

For fixed m ≥ 1, we study the number of weak left-to-right maxima which occur exactly m times in words whose letters satisfy a geometric distribution. First, we find the generating function and two exact expressions for the mean. Thereafter we use Rice’s integrals to derive an asymptotic formula as n tends to infinity for the average in random geom...

We consider the bargraph representation of geometrically distributed words, which we use to define the water capacity of such words. We first find a bivariate capacity generating function for all geometrically distributed words, from which we compute the generating function for the mean capacity. Thereafter, by making extensive use of Rice’s method...

We use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j . We also provide the asymptotics for compositions and words which have no alphabetic points,...

In a Dyck path a peak which is (weakly) higher than all the preceding peaks is called a strict (weak) left to right maximum. We obtain explicit generating functions for both weak and strict left to right maxima in Dyck paths. The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and...

We consider compositions of n represented as bargraphs and subject these to repeated impulses which start from the left at the top level and destroy horizontally connected parts. This is repeated while moving to the right first and then downwards to the next row and the statistic of interest is the number of impulses needed to annihilate the whole...

С помощью производящих функций получены оценки наличия (или отсутствия) разделяющих точек в разбиениях и словах. Разделяющая точка - это такое значение $j$, что все значения слева от него не больше $j$, а все значения справа от него не меньше $j$. Получены также асимптотические формулы для чисел разбиений и слов без разделяющих точек, когда размер...

In this paper, we consider a new statistic on the set S_n of permutations of length n which records the number of vertical edges within their bargraph representations. Indeed, we determine the distribution of this statistic on a class of multi-permutations having S_n as a subset. We compute an explicit formula for the total number of vertical edges...

We provide a particular measure for the degree to which an arbitrary composition deviates from increasing sorted order. The application of such a measure to the transport industry is given in the introduction. In order to obtain this measure, we define a statistic called the number of pushes in an arbitrary composition (which is required to produce...

A composition of a positive integer n is a representation of n as an ordered sum of positive integers a_1 + a_2 + · · · + a_m = n. There are 2^(n−1) unrestricted compositions of n, which can be classified according to the number of inversions they contain. An inversion in a composition is a pair of summands {a_i, a_j} for which i < j and a_i > a_j....

In this paper we consider compositions of n as bargraphs. The depth of a cell inside this graphical representation is the minimum number of horizontal and/or vertical unit steps that are needed to exit to the outside. The depth of the composition is the maximum depth over all cells of the composition. We use finite automata to study the generating...

A partition π of a set S is a collection B1 , B2 , …, B k of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1 , B2 , …, B k are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min B k . A partition...

We define the notion of capacity, the ability to contain water, for Dyck paths of semi-length n. Initially the Dyck paths attain a maximum height h and by summing over h the capacity generating function for all Dyck paths is obtained. Thereafter, we obtain the average Dyck path capacity generating function and finally an asymptotic expression for t...

We define \(P_{r}(q)\) to be the generating function which counts the total number of distinct (sequential) r-tuples in partitions of n and \(Q_r(q,u)\) to be the corresponding bivariate generating function where u tracks the number of distinct r-tuples. These statistics generalise the number of distinct parts in a partition. In the early part of t...

A composition of the positive integer n is a representation of n as an ordered sum of positive integers (Formula presented.) There are (Formula presented.) unrestricted compositions of n, which can be sorted according to the number of inversions they contain. (An inversion in a composition is a pair of summands (Formula presented.) for which (Formu...

Compositions of n are finite sequences of positive integers Such that σ1 + σ2 + · · · + σk = n. The σ's are called parts. We can represent a composition as a bargraph where the parts of the composition are represented by the columns and the height of each column corresponds to the size of the corresponding part. We consider the inner site-perimeter...

We study compositions (ordered partitions) of n. More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size s × s. We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition...

A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($\Gamma_1, \Gamma_2, \dots, \Gamma_n$) where $\P(\Gamma_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we direct...

A nonempty word w of finite length over the alphabet of positive integers is a Stirling word if for each letter i in w all entries between two consecutive occurrences of i (if these exist) are larger or equal to i. We derive an exact and also an asymptotic formula for the probability that a random geometrically distributed word of length n is a Sti...

We define [k]={1,2,…,k} to be a (totally ordered) alphabet on k letters. A wordw of length n on the alphabet [k] is an element of [k]ⁿ. A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the x-axis) in which the height of the ith column equals the size of the ith part of the word. Thus these bargrap...

We define [k] = {1, 2, 3, . . . , k} to be a (totally ordered) alphabet on k letters. A word w of length n on the alphabet [k] is an element of [k]^n. A word can be represented by a bargraph which is a family of column-convex polyominoes whose lower edge lies on the x-axis and in which the height of the i-th column in the bargraph equals the size o...

We study descents from maximal elements in samples of geometric random variables and consider two different averages for this statistic. We then compare the asymptotics of these averages as the number of parts in the samples tends to infinity, and also find an asymptotic expression for the mean of the greatest descent after a maximum value in such...

A word over an alphabet [k] can be represented by a bargraph, where the height of the i-th column is the size of the i-th part. If North is in the direction of the positive y-axis and East is in the direction of the positive x-axis, a light source projects parallel rays from the North-West direction, at an angle of 45 degrees to the y-axis. These r...

We study descents from maximal elements in samples of geometric random variables and consider two different averages for this statistic. We then compare the asymptotics of these averages as the number of parts in the samples tends to infinity, and also find an asymptotic expression for the mean of the greatest descent after a maximum value in such...

Bargraphs are lattice paths in N0² with three allowed types of steps; up (0, 1), down (0, −1) and horizontal (1, 0). They start at the origin with an up step and terminate immediately upon return to the x-axis. A wall of size r is a maximal sequence of r adjacent up steps. In this paper we develop the generating function for the total number of wal...

In this paper we study the largest parts in integer partitions according to multiplicities and part sizes. Firstly we investigate various properties of the multiplicities of the largest parts. We then consider the sum of the $m$ largest parts - first as distinct parts and then including multiplicities. Finally, we find the generating function for t...

Bargraphs are non-intersecting lattice paths in with 3 allowed types of steps; up (0, 1), down (0, −1) and horizontal (1, 0). They start at the origin with an up step and terminate immediately upon return to the x-axis. We unify the study of integer compositions (ordered partitions) with that of bargraph lattice paths by obtaining a single generati...

In this paper, compositions of n are studied. These are sequences of positive integers (Formula Presented) whose sum is n. We define a maximum to be a part which is greater than or equal to all other parts. We investigate the size of the descents immediately following any maximum and we focus particularly on the largest and average of these, obtain...

Bargraphs are specific polyominoes or lattice paths in . They start at the origin and end on the -axis. The allowed steps are the up step , the down step and the horizontal step . There are a few restrictions: the first step has to be an up step and the horizontal steps must all lie above the -axis. An up step cannot follow a down step and vice ver...

Bargraphs are lattice paths in N-0(2), which start at the origin and terminate immediately upon return to the x-axis. The allowed steps are the up step (0,1), the down step (0,-1) and the horizontal step (1,0). The first step is an up step and the horizontal steps must all lie above the x-axis. An up step cannot follow a down step and vice versa. I...

We consider samples of geometric random variables and find the average size of the descent after the first and last maximal values. These are asymptotically but not exactly equal, with the descent after the last maximum being slightly larger than that after the first. Thereafter we calculate the probability that the descent after the last maximum i...

A bargraph is a lattice path in with three allowed steps: the up step , the down step and the horizontal step . It starts at the origin with an up step and terminates as soon as it intersects the -axis again. A down step cannot follow an up step and vice versa. The height of a bargraph is the maximum coordinate attained by the graph. The width is t...

The coupon collector's problem is a classical problem usually solved by summing independent but not identically distributed geometric random variables. These geometric random variables represent waiting times for the j-th coupon for j = 1, 2, · · ·. In this paper we address the question about the longest of these waiting times. In particular, we ev...

The main result of this paper is the generalization and proof of a conjecture by Gould and Quaintance on the asymptotic behavior of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S , i.e., collections B1,B2,…,BkB1,B2,…,Bk of non-empty disjoint subse...

Combinatorics International audience We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ1+σ2+...+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functi...

A partition of an integer n is a representation n = a1 + a2 + ⋯ + ak, with integer parts a1 ≥ a2 ≥ ⋯ ≥ ak ≥ 1. The Durfee square is the largest square of points in the graphical representation of a partition. We consider generating functions for the sum of areas of the Durfee squares for various different classes of partitions of n. As a consequenc...

Let k, n be integers such that 1≤k≤n. Say that two compositions of n, into k parts are related if they differ only by an element of D k (the dihedral group on a set of k elements), that is, if they differ by a cyclic shift (translation) or by reversal of the parts (reflection). This defines an equivalence relation on the set of such compositions. L...

Combinatorics International audience In this paper, we consider random words ω1ω2ω3⋯ωn of length n, where the letters ωi ∈ℕ are independently generated with a geometric probability such that Pωi=k=pqk-1 where p+q=1 . We have a descent at position i whenever ωi+1 < ωi. The size of such a descent is ωi-ωi+1 and the descent variation is the sum of all...

By analogy with recent work of G. E. Andrews [The theory of partitions. Cambridge: Cambridge University Press (1998; Zbl 0996.11002); J. Reine Angew. Math. 624, 133–142 (2008; Zbl 1153.11053)] on smallest parts in partitions of integers, we consider smallest parts in compositions (ordered partitions) of integers. In particular, we study the number...

We prove some results about the coefficients r n of ∏ i≥0 (1+q 3i 2 +3i+1 ). These coefficients count the number of a special type of partitions of n, namely totally symmetric plane partitions with self conjugate main diagonal. In particular, we prove the conjecture that n=860 is the largest n such that r n =0.

In this paper, we consider random words ω 1 ω 2 ω 3 ⋯ω n of length n, where the letters ω i ∈ℕ are independently generated with a geometric probability such that ℙ{ω i =k}=pq k-1 where p+q=1. We have a descent at position i whenever ω i+1 <ω i . The size of such a descent is ω i -ω i+1 and the descent variation is the sum of all the descent sizes f...

We study samples T = (T1.....Tn) of length n where the letters Fi are independently generated according to the geometric distribution F(Fj = i) = pqi-1, for 1≤ j ≤n, with p + q=1 and 0<p<1. An up-smooth sample F is a sample such that F i+1-Fi ≤1. We find generating functions for the probability that a sample of n geometric variables is up-smooth, w...

We study permutations of the set [n] = {1,2,⋯,n} written in cycle notation, for which eiich cycle forms an increasing or decreasing interval of positive integers. More generally, permutations whose cycle elements form arithmetic progressions are considered. We also investigate the class of generalised interval permutations, where each cycle can be...

We consider words over the alphabet [k] = {1, 2, . . . , k}, k ≥ 2. For a fixed nonnegative integer p, a p-succession in a word w 1w 2 . . . w n consists of two consecutive letters of the form (w i , w i + p), i = 1, 2, . . . , n − 1. We analyze words with respect to a given number of contained p-successions. First we find the mean and variance of...

A composition σ = a1a2 . . . am of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if ai > aj (aiaj) for all j = 1,2, . . . , i 1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well...

Let b⩾2b⩾2 be a fixed positive integer and let S(n)S(n) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of S(n)S(n) in base b is at least c0logn/(loglogn), where c0c0 is some positive constant depending on b and on the sequence of binomial sums. Our results include middle binomial coefficients (2nn) a...

The general field of additive number theory considers questions concerning representations of a given positive integer n as a sum of other integers. In particular, partitions treat the sums as unordered combinatorial objects, and compositions treat the sums as ordered. Sometimes the sums are restricted, so that, for example, the summands are distin...

In this paper we consider the absolute variation statistics of a composition σ = σ1 ··· σm of n which is a measure of the sum of absolute differences between each consecutive pair of parts in a composition. This and some related statistics which we discuss, can also be interpreted within the context of bargraph polygons and bargraph walks of area n...

We prove that the Diophantine equation p1r1..p krk = has only finitely many positive integer solutions k, p1, .., pk, r1, .., rk, where p1, .., pk are distinct primes. If a positive integer n has prime factorization p1r1pkrk, then represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equati...

We consider samples of n geometric random variables !1 !2 . . . !n where P{!j = i} = pq i 1 , for 1 ≤ j ≤ n, with p + q = 1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exa...

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let $${\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}$$ denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n i...

A composition of the positive integer n is a representation of n as an ordered sum of positive integers n = a 1 + a 2 + · · · + am. It is well known that there are 2 n−1 compositions of n. An inversion in a composition is a pair of summands {a i , a j } for which i < j and a i > a j . The number of inversions of a composition is an indication of ho...

In this paper we find, asymptotically, the mean and variance for the largest missing value (part size) in a composition of an integer n. We go on to show that the probability that the largest missing value and the largest part of a composition differ by one is relatively high and we find the mean for the average largest value in compositions that h...

A composition of a positive integer n, ? = ?1?2 � � � ?N, where ?1 ?2 � � � ?N = n, is said to be smooth if it contains no pair of adjacent letters with difference greater than 1. A smooth composition ? is called cyclic if in addition it satisfies |?1 ? ?N | ? 1. In this paper we study the problem of enumerating the smooth compositions of n with pa...

A partition of [n] = {1,2,...,n} is a decomposition of [n] into nonempty subsets called blocks. We will make use of the canonical representation of a partition as a word over a finite alphabet, known as a restricted growth function. An element ai in such a word π is a strong (weak) record if ai> aj (ai � aj) for all j = 1,2,...,i −1. Furthermore, t...

We compute the asymptotic probability that two randomly selected compositions of n into parts equal to a or b have the same number of parts. In addition, we provide bijections in the case of parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets. Explicit algebraic generating functions and asymptotic probabil...

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We prove some theorems concerning.

A word σ = σx · · · σnover the alphabet [k] = {1, 2,..., k} is said to be a staircase if there are no two adjacent letters with difference greater than 1. A word σ is said to be staircase-cyclic if it is a staircase word and in addition satisfies |σn - σ| ≤ 1. We find the explicit generating functions for the number of staircase words and staircase...

International audience We investigate the probability that a random composition (ordered partition) of the positive integer $n$ has no parts occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{`forbidden set'}$ $A$ of multiplicities. This probability is also studied in the related case of samples $\Gamma =(\Gamma_1,\Gamma_...

We consider words or strings of characters a1a2a3…an of length n, where the letters ai ∈ N are independently generated with a geometric probabilityP{X = k} = pqk−1 where p + q = 1.Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if ai+1 ≥ ai + d. We study the average position, initial height and end height of th...

We consider samples of n geometric random variables (Γ 1 ,Γ 2 ,⋯Γ n ) where ℙ{Γ j =i}=pq i-1 , for 1≤j≤n, with p+q=1. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we compute the first two (factorial) moments. The asymp...

A partition of an integer n is a representation n=a 1+a 2+⋅⋅⋅+a k , with integer parts 1≤a 1≤a 2≤…≤a k . For any fixed positive integerp, a p-succession in a partition is defined to be a pair of adjacent parts such that a i+1−a i =p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total n...

Combinatorics International audience A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of t...

International audience A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions...

The aim of this paper is to study analytical and combinatorial methods to solve a special type of recurrence relation with two indices. It is shown that the recurrence relation enumerates a natural combinatorial object called a plane composition. In addition, further interesting recurrence relations arise in the study of statistics for these plane...

A word $\sigma=\sigma_1...\sigma_n$ over the alphabet $[k]=\{1,2,...,k\}$ is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word $\sigma$ is said to be {\em smooth cyclic} if it is a smooth word and in addition satisfies $|\sigma_n-\sigma_1|\le 1$. We find the explicit generating functions for the num...

Given integers m≥2, r≥2, let q m (n), q 0 (m) (n), b r (m) (n) denote the number of m-colored partitions of n into distinct parts, distinct odd parts, and parts not divisible by r, respectively. We obtain recurrences for each of the above-mentioned types of partition functions.

A partition of a positive integer n is a finite sequence of positive integers a 1 , a 2 , . . ., a k such that a 1 + a 2 +ċ ċ ċ+ a k = n and a i +1 ≥ a i for all i . Let d be a fixed positive integer. We say that we have an ascent of size d or more if a i +1 ≥ a i + d . We determine the mean, the variance and the limiting distribution of the number...

For words of length n, generated by independent geometric random variables, we consider the mean and variance, and thereafter the distribution of the number of runs of equal letters in the words. In addition, we consider the mean length of a run as well as the length of the longest run over all words of length n.

The Fibonacci number of a graph is the number of independent vertex subsets. In this paper, we investigate trees with large Fibonacci number. In particular, we show that all trees with n edges and Fibonacci number >2n-1+5 have diameter ⩽4 and determine the order of these trees with respect to their Fibonacci numbers. Furthermore, it is shown that t...

For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the last descent in the word. In addition we compute the average initial and end height of the last descent in a random permutation of n letters.

International audience We consider samples of n geometric random variables $(Γ _1, Γ _2, \dots Γ _n)$ where $\mathbb{P}\{Γ _j=i\}=pq^{i-1}$, for $1≤j ≤n$, with $p+q=1$. The parameter we study is the position of the first occurrence of the maximum value in a such a sample. We derive a probability generating function for this position with which we c...

We study several statistics for integer partitions: for a random partition of an integer n we consider the average size of the smallest gap (missing part size), the multiplicity of the largest part, and the largest repeated part size. Furthermore, we estimate the number of gap-free partitions of n.

We study the gaps or missing part sizes in partition of integers. For a random partition of an integer n we consider the average sizes of the largest gap and the total number of gaps. We show that the largest gap grows asymptotically at the same rate as the largest part in a partition and that the number of gaps grows with order √n.

For words of length n, generated by independent geometric random variables, we consider the average and variance of the number of distinct values (= letters) that occur in the word. We then generalise this to the number of values which occur at least b times in the word. 1.

International audience We investigate the probability that a sample $\Gamma=(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ of independent, identically distributed random variables with a geometric distribution has no elements occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{'forbidden set'}$ $A$ of multiplicities. Specific choice...

International audience A composition of a positive integer $n$ is a finite sequence of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+ \cdots +a_k=n$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more at position $i$, if $a_{i+1}\geq a_i+d$. We study the average position, initial height and end...

We study the number of ways of writing a positive integer n as a product of integer factors greater than one. We survey methods from the literature for enumerating and also generating lists of such factorizations for a given number n. In addition, we consider the same questions with respect to factorizations that satisfy constraints, such as having...

International audience For words of length n, generated by independent geometric random variables, we study the average initial and end heights of the first descent in the word. In addition we compute the average initial and end height of the first descent for a random permutation of n letters.

We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutativ...

We consider compositions or ordered partitions of the natural number n for which the largest (resp. smallest) summand occurs in the first position of the composition.

We study the asymptotic probability that a random composition of an integer n is gap-free, that is, that the sizes of parts in the composition form an interval. We show that this problem is closely related to the study of the probability that a sample of independent, identically distributed random variables with a geometric distribution is likewise...

We consider the totel number of parts in partitions of the natural number n, and derive identities relating this function to the number of partitions and to other familiar number theoretic functions. The total number of parts in partitions with distinct parts and partitions with other restrictions is also considered.