These theories introduce basic concepts and proofs about discrete summation: shifts, formal summation, falling factorials and stirling numbers. As proof of concept, a simple summation conversion is provided. 1 Stirling numbers of first and second kind theory Stirling imports Main begin 1.1 Stirling numbers of the second kind fun Stirling :: nat ⇒ nat ⇒ nat where Stirling 0 0 = 1 | Stirling 0 (S...

We classify k-Stirling permutations avoiding a set of ordered patterns of length three according to Wilf-equivalence. Moreover, we derive enumeration formulæ for all of the classes using a variety of techniques such as the kernel method, a bijection related to a classical result of Simion and Schmidt, and also structural decompositions of k-Stirling permutations via the so-called block decompos...

The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced by Everitt et al. (2002) and (2007) in the spectral theory. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obta...

1Occupational and Environmental Health Research Group, Centre for Public Health and Population Health Research, University of Stirling, Scotland, Stirling FK9 4LA, UK 2 Institute of Aquaculture, University of Stirling, Scotland, Stirling FK9 4LA, UK 3Stirling Management School, University of Stirling, Scotland, Stirling FK9 4LA, UK 4 Institute of Health and Wellbeing, University of Glasgow, Sco...

A permutation σ of a multiset is called Stirling permutation if σ(s) ≥ σ(i) as soon as σ(i) = σ(j) and i < s < j. In our paper we study Stirling polynomials that arise in the generating function for descent statistics on Stirling permutations of any multiset. We develop generalizations of the classical Stirling numbers and present their combinatorial interpretations. Particularly, we apply the ...

In this paper, we establish several properties of the unified generalized Stirling numbers of the first kind, and the Jacobi-Stirling numbers of the first kind, by means of the convolution principle of sequences. Obtained results include generalized Vandermonde convolution for the unified generalized Stirling numbers of the first kind, triangular recurrence relation for general Stirling-type nu...

and Belief-Based Language Differentiate Joking, Pretending, and Literal Toddler-Directed Speech Elena Hoicka ([email protected]) Psychology Department, University of Stirling, Stirling, FK9 4LA, UK Ruth Campbell ([email protected]) Psychology Department, University of Stirling, Stirling, FK9 4LA, UK

For any prime p we establish a congruence of Kummer type for the NN orlund polynomial B (x) n , and determine the highest power of p which divides the coeecient denominators of the NN orlund polynomial. This improves the result of 4] where only an upper bound was proven. We deduce a simple formula for the least common denominator of the coeecients. Applications are made for the Stirling polynom...

The left-truncated generalized Poisson distribution belongs to the family of the modified power series distributions. Using sufficiency and completeness of £ ^ x . (minx , , $ 3 x 0 ) respectively, when the truncation point is known (resp. unknown), the minimum variance unbiased (M.V.U) estimator for certain functions of the parameter $ (resp. 0, r) involved in these distributions can be obtain...

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, k-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [17]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generali...