The object of this paper is to present several classes of linear and bilateral generating relations by employing operational techniques, which reduces as a special case of (known or new) bilateral generating relations.At last some of generating functions associated with stirling number of second kind are also discussed. Mathematics Subject Classification(2010): Primary 42C05, Secondary 33C45.

We define the Sheffer group of all Sheffer-type polynomials and prove the isomorphism between the Sheffer group and the Riordan group. An equivalence of the Riordan array pair and generalized Stirling number pair is also presented. Finally, we discuss a higher dimensional extension of Riordan array pairs. AMS Subject Classification: 05A15, 11B73, 11B83, 13F25, 41A58

We study Stirling permutations defined by Gessel and Stanley in [4]. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents, and other, equidistributed statistics on these objects converge to a normal distribution.

A superdiagonal composition is one in which the i-th part or summand of size greater than equal to i. In this paper, we study number palindromic compositions and colored compositions. particular, give generating functions explicit combinatorial formulas involving binomial coefficients Stirling numbers first kind.

Thalidomide (a-[N-phthalmido]-glutarimide) was synthesized in 1954, in what was then West Germany, by Chemie Grünenthal under the brand name of Contergan and was subsequently licensed in 46 other countries worldwide, covering all continents. It is an odorless white crystalline compound with low solubility in water (McBride, ’77; Stirling et al., ’97). A number of pharmaceutical companies manufa...

Correspondence: [email protected] Computing and Mathematics, University of Stirling, FK9 4LA Stirling, UK Abstract Background: Mixed Order Hyper Networks (MOHNs) are a type of neural network in which the interactions between inputs are modelled explicitly by weights that can connect any number of neurons. Such networks have a human readability that networks with hidden units lack. They can be u...

We characterise the permutations π such that the elements in the closed lower Bruhat interval [id, π] of the symmetric group correspond to nontaking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id, π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner. Our characterisation connects the Poincaré p...

In this work, we propose a new statistical randomness test, the Saturation Point Test, which can be applied to integer sequences as well as binary sequences and is designed to increase the number of tests for short sequences. The subject of Saturation Point Test is the index of integer, denoted by SP , where all possible integers occur in the given sequence. We evaluate the probability Pr(SP = ...

A number of commentators have recently identified a 'uniquely Scottish mode of thought in economics', or a particular 'Scottish political economy tradition'. In this paper the concept of tradition, defined as those features common to practitioners over a relatively long period, is investigated in relation to 'school', 'paradigm' and 'research programme'. An application of the concept to the Sco...

We study three similar bijections on set partitions. The first gives a bijective proof of the equivalence of two statistics with a q-Stirling distribution, Milne’s statistic and the intertwining number. The second proves the equivalence of a multivariate block size distribution to a covering statistic. The third demonstrates equivalence of the number of all set partitions up to a given size to ...