Using a combinatorial approach, we introduce the translated Whitney numbers. This seems to be more natural than to write a product of anarithmetical progression in terms of a power variable and conversely. We also extend our ideas to translated r-Whitney numbers of both kinds and to translated Whitney-Lah numbers.

The complete and elementary symmetric functions are specializations of Schur functions. In this paper, we use this fact to give two identities for the complete and elementary symmetric functions. This result can be used to proving and discovering some algebraic identities involving rWhitney and other special numbers.

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

Let Tn denote a tree of order n and Wm a wheel of order m + 1. Baskoro et al. conjectured in [2] that if Tn is not a star, then R(Tn,Wm) = 2n − 1 for m ≥ 6 even and n ≥ m− 1. We disprove the Conjecture in [6]. In this paper, we determine R(Tn,W6) for n ≤ 8 which is the first step for us to determine R(Tn,W6) for any tree Tn.

The (q, r)-Whitney numbers were recently defined in terms of the q-Boson operators, and several combinatorial properties which appear to be q-analogues of similar properties were studied. In this paper, we obtain elementary and complete symmetric polynomial forms for the (q, r)-Whitney numbers, and give combinatorial interpretations in the context of A-tableaux. We also obtain convolution-type ...

Given non-negative integers r, s, and t, an [r, s, t]-coloring of a graph G = (V (G), E(G)) is a mapping c from V (G)∪E(G) to the color set {0, 1, . . . , k− 1}, k ∈ N, such that |c(vi)− c(vj)| ≥ r for every two adjacent vertices vi, vj , |c(ei)− c(ej)| ≥ s for every two adjacent edges ei, ej , and |c(vi)− c(ej)| ≥ t for all pairs of incident vertices and edges, respectively. The [r, s, t]-chro...

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q; r; w)Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q...

The Stirling number of the second kind {k} counts the number of ways to partition a set of n labeled balls into k non-empty unlabeled cells. We extend this problem and give a new statement of the r-Stirling numbers of the second kind and r-Bell numbers. We also introduce the r-mixed Stirling number of the second kind and r-mixed Bell numbers. As an application of our results we obtain a formula...

Simulation techniques are a core component of scientific computing and, more specifically, computational statistics. All simulation methods—Monte Carlo methods, bootstrapping, estimation by simulation to name but a few—rely on ‘computer-generated randomness’ (more on this below). In practice, this means sequences of random numbers. Generating ‘good’ (for a suitable metric) random numbers is the...

Marked mesh patterns are a very general type of permutation pattern. We examine a particular marked mesh pattern originally defined by Kitaev and Remmel, and show that its generating function is described by the r-Stirling numbers. We examine some ramifications of various properties of the r-Stirling numbers for this generating function, and find (seemingly new) formulas for the r-Stirling numb...