On Bell numbers of type $D$

TL;DR

This paper defines Bell numbers of type $D$, $D(n)$, as sums of type $D$ Stirling numbers over signed set partitions, and derives a recurrence, an exponential generating function $D(x)=e^{\frac{e^{2x}-1}{2}}(e^x-x)$, and an explicit formula $D(n)=e^{-1/2}\sum_{r\ge 0}\frac{1}{2^r r!}\left[(2r+1)^n - n(2r)^{n-1}\right]$. It also situates $D(n)$ within the landscape of classical and type-$B$ Bell numbers, via relations among $S(n,k)$, $S_B(n,k)$, and $S_D(n,k)$, and outlines several open questions linking $D$-type partitions to derangements of type $D$, $G(m,p,n)$-partitions, and flattened Stirling permutations. The results provide concrete computational tools and a framework for further generalizations in signed partition theory. The work broadens the combinatorial understanding of partition analogues beyond the classical Bell numbers.

Abstract

In this paper, we will introduce Bell numbers $D(n)$ of type $D$ as an analogue to the classical Bell numbers related to all the partitions of the set $[n]$. Then based on a signed set partition of type $D$, we will construct the recurrence relations of Bell numbers $D(n)$. In addition, we deduce the exponential generating function for $D(n)$. Finally, we will provide an explicit formula for $D(n)$.

Theorems & Definitions (12)

• Example 1.1
• Definition 1.2: br1'
• Proposition 1.3
• proof
• Definition 2.1
• Theorem 2.2
• proof
• Example 2.3
• Theorem 2.4
• proof