Prefab posets` Whitney numbers

TL;DR

This paper introduces a natural partial order on certain finite sets called cobweb prefabs, calculates their Whitney numbers, and explores connections to Fibonacci triad sequences and binomial-like coefficients.

Contribution

It provides explicit formulas for Whitney numbers of cobweb prefab posets and extends relations between binomial coefficients and Fibonacci sequences.

Findings

Calculated Whitney numbers form Stirling-like triangular arrays.

Established connections between prefab posets and Fibonacci triad sequences.

Extended binomial coefficient relations to Fibonacci-like sequences.

Abstract

We introduce a natural partial order in structurally natural finite subsets the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers` triangular array are then calculated and the explicit formula for them is provided. Next - in the second construction - we endow the set sums of prefabiants with such an another partial order that their their bell like numbers include fibonacci triad sequences introduced recently by the present author in order to extend famous relation between binomial newton coefficients and fibonacci numbers onto the infinity of their relatives among which there are also the fibonacci triad sequences and binomial-like coefficients (incidence coefficients included).