In the present paper, we consider two kinds of statistics “number of usegments” and “number of internal u-segments” in Dyck paths. More precisely, using Lagrange inversion formula we present the generating function for the number of Dyck paths according to semilength and our new statistics by the partial Bell polynomials, namely, ∑ D∈Dn ∏ i≥1 t αi(D) i = n ∑ i=1 1 (n− i+ 1)!n,i ( 1!t1, 2!t2, · ...

In this paper, by introducing degenerate Fubini-type polynomials, with the help of Faà di Bruno formula and some properties partial Bell authors provide several new explicit formulas recurrence relations for polynomials numbers, associate newly defined Apostol–Bernoulli Apostol–Euler order α. These results enable one to present additional special numbers.

Letting Bn,r be the n-th r-Bell polynomial, it is well known that Bn(x) admits specific integer coordinates in the two bases {x}i and {xBi(x)}i according to, respectively, the Stirling numbers and the binomial coefficients. Our aim is to prove that the sequences Bn+m,r(x) and Bn,r+s(x) admit a binomial recurrence coefficient in different bases of the Q-vector space formed by polynomials of Q[X].

We derive a formula for p(n) (the number of partitions n) in terms the partial Bell polynomials using Faà di Bruno’s and Euler’s pentagonal theorem.

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (pp − 1)/(p− 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p...

We derive some additional results on the Bienyamé-Galton-Watson branching process with θ−linear fractional branching mechanism, as studied in [16]. This includes: the explicit expression of the limit laws in both the sub-critical cases and the super-critical cases with finite mean, the long-run behavior of the population size in the critical case, limit laws in the super-critical cases with inf...

A hierarchy is a distribution of elements on levels. For a given number of elements a certain hierarchy can be arrangend in a finite number of ways. We provide a counting formula for certain types of hierarchies. Mainly, this formula is a sum over the integer partitions of the number of elements. Furthermore, several basic combinatorical numbers like the Stirling numbers of the second kind, the...

This entry defines the Bell numbers [1] as the cardinality of set partitions for a carrier set of given size, and derives Spivey’s generalized recurrence relation for Bell numbers [2] following his elegant and intuitive combinatorial proof. As the set construction for the combinatorial proof requires construction of three intermediate structures, the main difficulty of the formalization is hand...

The Grishukhin inequality is a facet of the cut polytope CUT7 of the complete graph K7, for which no natural generalization to a family of inequalities has previously been found. On the other hand, the Imm22 Bell inequalities of quantum information theory, found by Collins and Gisin, can be seen as valid inequalities of the cut polytope CUT¤(K1,m,m) of the complete tripartite graph K1,m,m. They...

In the present paper, we define generalized Kwang-Wu Chen matrix. Basic properties of this generalization, such as explicit formulas and generating functions are presented. Moreover, focus on a new class Fubini polynomials. Then discuss their relationship with other polynomials Fubini, Bell, Eulerian Frobenius-Euler We have also investigated some basic related to degenerate