New type Stirling like numbers - an email style letter

TL;DR

This paper introduces new Stirling-like numbers derived from generalized Fibonacci cobweb posets and explores their combinatorial properties and connections to Bell-like numbers and partial orders.

Contribution

It extends Fibonacci cobweb posets and prefab notions to define new Stirling-like numbers with associated partial orders and Bell-like sequences.

Findings

Introduction of new Stirling-like numbers from cobweb posets

Connection to Bell-like numbers and Fibonacci sequences

Development of partial orders on cobweb prefab sets

Abstract

The notion of the Fibonacci cobweb poset from [1] has been naturally extended to any admissible sequence $F$ in [2] where it was also recognized that the celebrated prefab notion of Bender and Goldman [3] - (see also [4,5]) - admits such an extension so as to encompass the new type combinatorial objects from [2] as leading examples. Recently the present author had introduced also [6] two natural partial orders in there: one $\leq$ in grading-natural subsets of cobweb`s prefabs sets [2] and in the second proposal one endows the set sums of the so called "prefabiants" with such another partial order that one arrives at Bell-like numbers including Fibonacci triad sequences introduced by the present author in [7]. Here we quote the basic observations concerning the new type Stirling like numbers as they appear in [6]. For more on notation, Stirling like numbers of the first kind and for proofs - see [6].