Combinatorics of higher order degenerate r-deranged bell numbers with singletons

Technical Summary: Combinatorics of Higher Order Degenerate r-Deranged Bell Numbers with Singletons

Problem Statement
The paper addresses the combinatorial interpretation and structural properties of a new generalization of Bell numbers, termed "higher order degenerate r-deranged Bell numbers with singletons," denoted as $B_{n}^{r,x,\lambda}(\alpha, \beta, \gamma)$. While barred preferential arrangements (ordered set partitions with inserted bars) and $r$-deranged partitions (partitions where the first $r$ elements are in distinct cycles or are singletons) have been studied individually, this work seeks to unify these concepts. Specifically, it aims to define and interpret numbers that combine:

1. Degeneracy: Introduced via generalized Stirling numbers $S(n, k, \alpha, \beta, \gamma)$ involving parameters $\alpha, \beta, \gamma$.
2. Derangement constraints: Specifically $r$-derangements where the first $r$ elements of the set $[n+r]$ are singletons and must reside in different cycles.
3. Barred structures: The insertion of bars to create sections, generalized to "higher order" via a parameter $\lambda$.
4. Singletons: Explicitly handling the first $r$ elements as singleton blocks.

The central question posed is how to provide a rigorous combinatorial interpretation for the generating function defined in Equation (11) of the paper, which generalizes existing $r$-deranged Bell numbers.

Methodology
The study employs generating functions and combinatorial constructions based on generalized Stirling numbers.

• Generalized Stirling Numbers: The authors utilize $S(n, k, \alpha, \beta, \gamma)$, defined via the expansion of factorial polynomials $(t|\alpha)_n$. These numbers generalize classical Stirling numbers, Carlitz degenerate numbers, and others, depending on the parameters $(\alpha, \beta, \gamma)$.
• Combinatorial Models: The paper constructs specific partition models involving "cells" and "compartments."
  • Cells: Blocks of a partition.
  • Compartments: Subdivisions within cells, subject to cyclic ordering and occupancy constraints (at most one element per compartment, with specific rules for $\alpha$-consecutive compartments).
  • Barred Preferential Arrangements: The insertion of $\lambda-1$ identical bars between linearly ordered blocks to induce sections.
• Derangement Constraints: The construction enforces that the first $r$ elements of the set $[n+r]$ form singleton blocks and that these singletons must appear in different cycles when the blocks are linearly ordered.
• Asymptotic Analysis: The paper applies a method developed in previous literature (cited as [21, 22]) involving formal power series and coefficients of the form $a(\alpha\delta, n)$ to derive asymptotic results for large $n$.

Key Contributions and Results

1. Definition of New Numbers:
   The paper defines the numbers $B_{n}^{r,x,\lambda}(\alpha, \beta, \gamma)$ via the generating function:
   $$\sum_{n=0}^{\infty} B_{n}^{r,x,\lambda}(\alpha, \beta, \gamma) \frac{t^n}{n!} = (1 + \alpha t)^{\gamma/\alpha} \frac{[x((1 + \alpha t)^{\beta/\alpha} - 1)]^{r\lambda} \exp(-\lambda[x((1 + \alpha t)^{\beta/\alpha} - 1)])}{[1 - x((1 + \alpha t)^{\beta/\alpha} - 1)]^{(r+1)\lambda}}$$
   This generalizes the standard $r$-deranged Bell numbers (recovered when $\alpha \to 0$, $\beta=1$, $\gamma=r$, $\lambda=1$, $x=1$).

2. Combinatorial Interpretation (The Case $\lambda=1$):
   For the degenerate case ($\lambda=1$), the authors define $(x, \alpha, \beta, 1, r)$-partitions. These are partitions of $[n+r]$ into $k+r+1$ blocks satisfying:

   • The first $r$ blocks are singletons formed by the first $r$ elements.
   • The remaining $k$ blocks are "like" blocks with $\beta$ labeled compartments.
   • The $(k+r+1)$-th block has $\gamma$ labeled compartments and may be empty.
   • The first $k+r$ blocks are linearly ordered such that the $r$ singletons are in different cycles (using standard form as reference).
   • The $k$ blocks are colored with $x$ available colors.
   • Specific occupancy rules apply to the compartments based on $\alpha$.
     The number of such partitions is given by $d_{k,r} x^k \beta^k S(n, k, \alpha, \beta, \gamma)$, where $d_{k,r}$ are $r$-derangement numbers.

3. Combinatorial Interpretation (General Case $\lambda$):
   For arbitrary $\lambda \in \mathbb{N}$, the authors introduce $(x, \alpha, \beta, \lambda, r)$-partitions. This extends the previous model by inserting $\lambda-1$ identical bars between the linearly ordered first $k+r$ blocks, creating a barred preferential arrangement with $\lambda+1$ sections.

   • Theorem 3.1 establishes a closed-form identity:
     $$B_{n}^{r,x,\lambda}(\alpha, \beta, \gamma) = \sum_{k=0}^{n} \binom{k+r+\lambda-1}{k+r} d_{k,r} x^k \beta^k S(n, k, \alpha, \beta, \gamma)$$
     This interprets the numbers as a sum over $k$ of the product of $r$-derangement counts, coloring factors, generalized Stirling numbers, and the number of ways to place bars.

4. Combinatorial Identities:

   • Theorem 3.2 provides a convolution identity expressing $B_{n}^{r,x,\lambda}$ as a sum over multinomial coefficients involving products of lower-order $B$ numbers and generalized falling factorials $(\gamma|\alpha)_{i_{\lambda+1}}$. This reflects the structure of distributing elements into $\lambda+1$ sections defined by the bars.
   • Theorem 3.3 relates the generalized Bell polynomials $\omega_{n+r}$ to the new numbers, distinguishing between "fixed" blocks and "permuted" blocks in the context of barred arrangements.

5. Asymptotic Results:
   Corollary 4.1 provides an asymptotic expansion for $B_{n}^{r,x,\lambda}(\alpha, \beta, \gamma)$ as $n \to \infty$ (under specific conditions on $\delta$). The result is expressed as a sum involving coefficients $W(n, f)$, which are derived from the partition of integers and the values of $B_{i,1}^{r,x}(\alpha, \beta, \gamma)$.

Significance and Claims
The paper claims to provide a unified combinatorial framework for a broad class of Bell-like numbers. By interpreting $B_{n}^{r,x,\lambda}(\alpha, \beta, \gamma)$ through the lens of barred preferential arrangements with specific derangement and singleton constraints, the study offers a concrete model for these abstract generating functions.

The authors state that their proposed combinatorial interpretation can be adapted to study other Bell-like polynomials and numbers, as the underlying generalized Stirling numbers $S(n, k, \alpha, \beta, \gamma)$ encompass many known combinatorial sequences (e.g., classical Stirling, Carlitz, Riordan, and Whitney numbers). The work is presented as a foundation for future research into the properties of these numbers, including the study of their roots and potential extensions to non-degenerate partial deranged Bell numbers. The paper does not claim experimental applications but rather theoretical unification and the derivation of new identities and asymptotic behaviors.