In this paper we extend the definition of Stirling numbers of the first kind by way of a special multiset. This results in a family of number triangles for which we show how to obtain ordinary generating functions for the rows and exponential generating functions for the columns. The latter are derived via a recursive process. We also indicate how to obtain formulas, in terms of factorials, gen...

In the paper, authors collect, discuss, and find out several connections, equivalences, closed-form formulas, combinatorial identities concerning partial Bell polynomials, falling factorials, rising extended binomial coefficients, Stirling numbers of first second kinds. These results are new, interesting, important, useful, applicable in number theory.

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the q-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized p, q-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definit...

Starting with divided di erences of binomial coe cients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Some particular calculations are done and several factorization results are proven and conjectured.

It is by now well known that the parities of the binomial coefficients show a fractal-like appearance when plotted in the x-y plane. Similarly, if f(n,k) is some counting sequence and/? is a prime, we can plot an asterisk at (//,&) iff(n,k) # 0 (mod/?), and a blank otherwise, to get other complex, and often interesting, patterns. For the ordinary and Gaussian binomial coefficients and for the S...

wild and commercial bees and pesticide use Adam Kleczkowski∗ Computing Science and Mathematics, School of Natural Sciences, University of Stirling, UK Ciaran Ellis Biological and Environmental Studies, School of Natural Sciences, University of Stirling, UK Dave Goulson School of Life Sciences, University of Sussex, UK Nick Hanley Dept. of Geography and Sustainable Development, University of St ...

J. S. Tyrrell, C Morton, S. M. Campbell and A. Curnow Clinical Photobiology, European Centre of Environment and Human Health, Peninsula Medical School, University of Exeter, Knowledge Spa, Royal Cornwall Hospital, Treliske, Truro, Cornwall, TR1 3HD, UK. 2 Stirling Royal Infirmary, Stirling, Scotland, FK8 2AU Corresponding author: Alison Curnow Email: [email protected] Telephone: 00 44 (0)...

a Balaton Limnological Institute, Hungarian Academy of Sciences Centre for Ecological Research, Klebelsberg K. u. 3, Tihany 8237, Hungary b University of Leicester, Centre for Landscape and Climate Research, Bennett Building, University Road, Leicester LE1 7RH, UK c Brockmann Consult, Max-Planck-Str. 2, 21502 Geesthacht, Germany d Biological and Environmental Sciences, School of Natural Science...

1 Laboratoire d’Ethologie Expérimentale et Comparée, Université Paris 13 Sorbonne Paris Cité, Villetaneuse, France, 2 Unité Transversale de Recherche Psychogenèse et Psychopathologie, EA4403, Université Paris 13 Sorbonne Paris Cité, Villetaneuse, France, 3 Centre des Sciences du Goût, Centre National de la Recherche Scientifique UMR 6265, Université de Bourgogne, Dijon, France, 4 Division of Ps...