Type-B analogue of Bell numbers using Rota's Umbral calculus approach

TL;DR

This paper extends classical Bell number properties to Coxeter type B partitions using Rota's umbral calculus, revealing new formulas and positivity results for the analogous numbers.

Contribution

It introduces a type-B analogue of Bell numbers and extends Rota's and Tanny's results to Coxeter type B set partitions.

Findings

Derived new formulas for type-B Bell numbers.

Established gamma-positivity for ordered type-B Bell numbers.

Extended Rota's and Tanny's results to Coxeter type B framework.

Abstract

Rota used the functional L to recover old properties and obtain some new formulas for the Bell numbers. Tanny used Rota's functional L and the celebrated Worpitzky identity to obtain some expression for the ordered Bell numbers, which can be seen as an evident to the fact that the ordered Bell numbers are gamma-positive. In this paper, we extend some of Rota's and Tanny's results to the framework of the set partitions of Coxeter type B.