A Combinatorial Approach to the Stirling Numbers of the First Kind with Higher Level

In this paper, we investigate a generalization of the classical Stirling numbers of the first kind by considering permutations over tuples with an extra condition on the minimal elements of the cycles. The main focus of this work is the analysis of combinatorial properties of these new objects. We give general combinatorial identities and some recurrence relations. We also show some connections with other sequences such as poly-Cauchy numbers with higher level and central factorial numbers. To obtain our results, we use pure combinatorial arguments and classical manipulations of formal power series.

A Parametric Type of Cauchy Polynomials with Higher Level

There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 is its analogue. In this paper, as an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level. We present their characteristic and combinatorial properties. By using recursions, we show some determinant expressions.

On the unsteady dynamics of synthetic leaves: Laboratory experiments using synchronized PIV and DIC

The unsteady 3D dynamics of various synthetic leaves and the induced turbulence are systematically studied experimentally for representative Cauchy numbers in a wind tunnel under nearly uniform incoming flows. Synchronized digital image correlation (DIC) and high-frame-rate particle image velocimetry (PIV) are employed to track the structure dynamics simultaneously and the surrounding flow field to uncover the fluid-solid interaction. A high-resolution six-axis load cell is also used to quantify the synthetic leaves' induced force and torque under various flows. The shapes of synthetic leaves inspected are representative of selected environments (e.g., calm to windy weather; tropical to temperate climate). The Cauchy number is set to resemble those observed in natural conditions. This presentation will discuss insights from synchronized PIV-DIC techniques on the synthetic leaves' distinct behavior and wake flow response. Particular emphasis is placed on characterizing flow instability and the leave shape's role in the motions and force. For this purpose, we inspected the instantaneous force and torque as well as their structure. We will also discuss the relationship between leave shapes with force and torque fluctuations linking them with the leaf motion obtained from DIC measurements. In particular, the results show that selected leaf shapes experience significantly larger and distinct force and torque fluctuations and larger pitch magnitude, as shown in Fig. 5. A shared monotonically decreasing trend of the nondimensional frequency (Strouhal number, St = fL/U) is evidenced for standard environmental conditions.

Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers

In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

A Note on the Degenerate Poly-Cauchy Polynomials and Numbers of the Second Kind

In this paper, we consider the degenerate Cauchy numbers of the second kind were defined by Kim (2015). By using modified polyexponential functions, first introduced by Kim-Kim (2019), we define the degenerate poly-Cauchy polynomials and numbers of the second kind and investigate some identities and relationship between various polynomials and the degenerate poly-Cauchy polynomials of the second kind. Using this as a basis of further research, we define the degenerate unipoly-Cauchy polynomials of the second kind and illustrate their important identities.