Wilf conjecture

TL;DR

This paper proves Wilf's conjecture regarding the non-zero alternating sum of Stirling numbers of the second kind for all but possibly one specific case related to weighted Motzkin numbers.

Contribution

The paper confirms Wilf's conjecture for all cases except potentially one, linking it to properties of weighted Motzkin numbers.

Findings

Wilf conjecture is true for all n > 2 except possibly one case.

The exception relates to properties of weighted Motzkin numbers.

Provides a proof connecting Stirling numbers and weighted Motzkin numbers.

Abstract

Let S(n,k) be the Stirling number of the second kind. Wilf conjectured that the alternating sum of S(n,k) for k from 0 to n is not zero for all n>2. In this paper, we prove that Wilf conjecture is true except at most one number with the properties of weighted Motzkin number.