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

Eli Bagno, David Garber

TL;DR

Some of Rota's and Tanny's results to the framework of the set partitions of Coxeter type B are extended and seen as an evident to the fact that the ordered Bell numbers are gamma-positive.

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.

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

TL;DR

Some of Rota's and Tanny's results to the framework of the set partitions of Coxeter type B are extended and seen as an evident to the fact that the ordered Bell numbers are gamma-positive.

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.

Paper Structure

This paper contains 6 sections, 6 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Set partitions and Bell numbers of type B
  3. Set partitions of type B
  4. Stirling numbers and Bell numbers of type B
  5. Type- B analogue of Rota's result
  6. Gamma-positivity of ordered Bell polynomial of type B

Key Result

Theorem 1.1

Let $n \geq 0$. Then: (1) $B_n = L(u^n)$, (2) $B_{n+1} = \sum\limits_{j=0}^n {n \choose j} B_j$, (3) $B_n = \frac{1}{e} \sum\limits_{j \geq 0} \frac{j^n}{j!}$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 3.1
  • proof
  • ...and 4 more