Derangement Numbers Research Articles

Some New Identities for Derangement Numbers

Some New Identities for Derangement Numbers

Identities between harmonic, hyperharmonic and Daehee numbers

In this paper, we present some identities relating the hyperharmonic, the Daehee and the derangement numbers, and we derive some nonlinear differential equations from the generating function of a hyperharmonic number. In addition, we use this differential equation to obtain some identities in which the hyperharmonic numbers and the Daehee numbers are involved.

The log-convexity of $r$-derangement numbers

This paper focuses on the log-convexity of the sequence $\{D_r(n)\}_{n\ge r}$ of $r$-derangement numbers, where $r\ge 2$ is a positive integer. We mainly prove that $\{D_2(n)\}_{n\ge 2}$ and $\{D_3(n)\}_{n\ge 7}$ are log-convex. In addition, we also show that $\{\sqrt {D_2(n)}\}_{n\ge 2}$ and $\{\sqrt [3]{D_3(n)}\}_{n\ge 7}$ are log-balanced.

A note on some identities of derangement polynomials

The problem of counting derangements was initiated by Pierre Rémond de Montmort in 1708 (see Carlitz in Fibonacci Q. 16(3):255–258, 1978, Clarke and Sved in Math. Mag. 66(5):299–303, 1993, Kim, Kim and Kwon in Adv. Stud. Contemp. Math. (Kyungshang) 28(1):1–11 2018. A derangement is a permutation that has no fixed points, and the derangement number d_{n} is the number of fixed-point-free permutations on an n element set. In this paper, we study the derangement polynomials and investigate some interesting properties which are related to derangement numbers. Also, we study two generalizations of derangement polynomials, namely higher-order and r-derangement polynomials, and show some relations between them. In addition, we express several special polynomials in terms of the higher-order derangement polynomials by using umbral calculus.

Application of Probabilistic Method on Daehee Sequences

In this paper, we investigate some combinatorial sequences based on Daehee and Changhee numbers and polynomials, then derive their moment representations in use of probabilistic method. We also provide identities related to Daehee numbers, derangement numbers, Cauchy numbers of the second kind, and Stirling numbers of the first kind.

A representation for derangement numbers in terms of a tridiagonal determinant

In the paper, the authors discover a representation for the derangement numbers in terms of a tridiagonal determinant. From the determinantal representation, the authors recover several identities and recurrence relations of the derangement numbers.

Closed forms for derangement numbers in terms of the Hessenberg determinants

In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.

Divisibility properties of the r-Bell numbers and polynomials

In the present article we extend several arithmetical results to a family of generalized Bell numbers called r-Bell numbers. In particular, we generalize some congruences such as Touchard's congruence, Sun–Zagier congruence for Bell and derangement numbers and polynomials, among others. We also describe the behavior of the 2-adic valuation of the r-Bell numbers.

The [formula omitted]-derangement numbers

The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, p-adic valuations, and diophantine problems.

The Monotonicity and Log-Behaviour of Some Functions Related to the Euler Gamma Function

AbstractThe aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

A recovery of two determinantal representations for derangement numbers

In the paper, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant.

A determinantal representation for derangement numbers

In the note, the authors find a representation for derangement numbers in terms of a tridiagonal determinant whose elements are the first few natural numbers.

Derangements on a Ferrers board

We study the derangement number on a Ferrers board B = (n × n) - λ with respect to an initial permutation M, that is, the number of permutations on B that share no common points with M. We prove that the derangement number is independent of M if and only if λ is of rectangular shape. We characterize the initial permutations that give the minimal and maximal derangement numbers for a general Ferrers board, and present enumerative results when λ is a rectangle.

Frame patterns in [formula omitted]-cycles

In this paper, we study the distribution of the number occurrences of the simplest frame pattern, called the μ pattern, in n-cycles. Given an n-cycle C, we say that a pair 〈i,j〉 matches the μ pattern if i<j and as we traverse around C in a clockwise direction starting at i and ending at j, we never encounter a k with i<k<j. We say that 〈i,j〉 is a trivial μ-match if j=i+1 and is a nontrivial μ-match if i+1<j. We say that an n-cycle C is incontractible if there is no i such that i+1 immediately follows i in C. We show that number of incontractible n-cycles in the symmetric group Sn is Dn−1 where Dn is the number of derangements in Sn. We show that number of n-cycles in Sn with exactly k nontrivial μ-matches can be expressed as a linear combination of binomial coefficients of the form (n−1i) where i≤2k+1. We also show that the generating function NTIn,μ(q) of q raised to the number of nontrivial μ-matches in C over all incontractible n-cycles in Sn is a new q-analogue of Dn−1 which is different from the q-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Schüzenberger and our polynomials in that the coefficient of the smallest power of q in NTI2k+1,μ(q) is the number of permutations in S2k+1 whose charge path is a Dyck path. Finally, we show that the coefficient of q(n−12)−k in NTIn,μ(q) is the number of partitions of k for sufficiently large n.

The Kurepa-Vandermonde matrices arising from Kurepa’s left factorial hypothesis

Kurepa?s (left factorial) hypothesis asserts that for each integer n ? 2 the greatest common divisor of !n := Pn-1?k=0 k! and n! is 2. It is known that Kurepa?s hypothesis is equivalent to p-1?k=0 (-1)k/k!?/ 0 (mod p) for each odd prime p, or equivalently, Sp-1 ?/ 0(modp) (i.e., Bp-1 ?/1(modp)) for each odd prime p, where Sp-1 and Bp-1 are the (p-1)th derangement number and the (p-1)th Bell number, respectively. Motivated by these two reformulations of Kurepa?s hypothesis and a congruence involving the Bell numbers and the derangement numbers established by Z.-W. Sun and D. Zagier [28, Theorem 1.1], here we give two ?matrix? formulations of Kurepa?s hypothesis over the field Fp, where p is any odd prime. The matrices Vp and Cp which are involved in these ?matrix? formulations of Kurepa?s hypothesis are the square (p-1)x(p-1) Vandermondelike matrices. Accordingly, Vp and Cp are called the Kurepa-Vandermonde matrices. Furthermore, for each odd prime p we determine det(Vp) and det(Cp) in the field Fp.

Infinitely log-monotonic combinatorial sequences

We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {an}n⩾0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {an+1/an}n⩾0 is log-concave. Furthermore, we prove that if a sequence {an}n⩾k is ratio log-concave, then the sequence {ann}n⩾k is strictly log-concave subject to an initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers Dn, we confirm a conjecture of Sun on the log-concavity of the sequence {Dnn}n⩾1.

Congruences on the Bell polynomials and the derangement polynomials

In this note, by the umbral calculus method, the Sun and Zagier congruences involving the Bell numbers and the derangement numbers are generalized to the polynomial cases. Some special congruences are also presented.

The Abel-Zeilberger Algorithm

By combining Abel's lemma on summation by parts with Zeilberger's algorithm, we give an algorithm, called the Abel-Zeilberger algorithm, to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, an identity of Andrews and Paule, and an identity of Calkin.

ON A CURIOUS PROPERTY OF BELL NUMBERS

AbstractIn this paper we derive congruences expressing Bell numbers and derangement numbers in terms of each other modulo any prime.

The ratio monotonicity of the [formula omitted]-derangement numbers

We introduce the notion of ratio monotonicity for polynomials with nonnegative coefficients, and we show that, for n ≥ 6 , the q -derangement numbers D n ( q ) are strictly ratio monotone except for the last term when n is even. This property implies the spiral property and log-concavity.