[Paper Review] Combinatorial approach of certain generalized Stirling numbers

Hacène Belbachir, Amine Belkhir+1 more|arXiv (Cornell University)|Nov 23, 2014

Advanced Mathematical Identities6 references3 citations

TL;DR

This paper presents a combinatorial interpretation and explicit formula for generalized Stirling numbers $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$, defined via a weighted distribution of $n$ elements into $k$ ordered non-empty lists with weights $\beta$ for list heads and $\alpha$ for other elements. Using inclusion-exclusion and symmetric functions, the authors derive recurrence relations, explicit expressions, and convolution identities, extending classical Stirling and Whitney numbers with new algebraic and combinatorial structure.

ABSTRACT

A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.

Motivation & Objective

• To provide a combinatorial interpretation for the generalized Stirling numbers $S(n,k;\alpha,\beta,0) = \genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$, extending classical Stirling and Whitney numbers.
• To derive an explicit formula for $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ using the inclusion-exclusion principle.
• To establish recurrence relations and symmetric function expressions for the generalized numbers.
• To prove new convolution identities involving multinomial and ordered partition structures.

Proposed method

• Define $\Omega_{n,k}$ as the set of weighted distributions of $n$ elements into $k$ ordered non-empty lists with head weight $\beta$, other elements weight $\alpha$, and first element weight $1$.
• Use the inclusion-exclusion principle to derive the explicit formula $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta} = \sum_{j=0}^{k} (-1)^{k-j} \binom{n}{j} \frac{(\beta j + \alpha)^{n-j}}{j!}$.
• Establish a recurrence relation: $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta} = \genfrac{\lfloor}{\rfloor}{0pt}{}{n-1}{k-1}^{\alpha,\beta} + (\alpha k + \beta (k-1)) \genfrac{\lfloor}{\rfloor}{0pt}{}{n-1}{k}^{\alpha,\beta}$.
• Express the numbers using elementary symmetric functions via generating functions and combinatorial weight assignments.
• Prove a multinomial convolution identity: $\binom{k}{k_1,\ldots,k_p} \genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta} = \sum_{l_1+\cdots+l_p=n} \binom{n}{l_1,\ldots,l_p} \genfrac{\lfloor}{\rfloor}{0pt}{}{l_1}{k_1}^{\alpha,\beta} \cdots \genfrac{\lfloor}{\rfloor}{0pt}{}{l_p}{k_p}^{\alpha,\beta}$.
• Derive a second convolution identity by analyzing the insertion of $s$ last elements into $k$ lists with weighted choices based on list positions and initial element retention.

Experimental results

Research questions

• RQ1 How can the generalized Stirling numbers $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ be interpreted combinatorially in terms of weighted list distributions?
• RQ2 What explicit closed-form expression can be derived for $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ using inclusion-exclusion?
• RQ3 How do recurrence relations for $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ generalize classical Stirling and Whitney number recurrences?
• RQ4 Can symmetric function theory be used to express $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ in a structured algebraic form?
• RQ5 What convolution identities govern the multiplicative and additive structure of these generalized numbers?

Key findings

• The explicit formula for $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ is derived via inclusion-exclusion: $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta} = \sum_{j=0}^{k} (-1)^{k-j} \binom{n}{j} \frac{(\beta j + \alpha)^{n-j}}{j!}$.
• A recurrence relation is established: $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta} = \genfrac{\lfloor}{\rfloor}{0pt}{}{n-1}{k-1}^{\alpha,\beta} + (\alpha k + \beta (k-1)) \genfrac{\lfloor}{\rfloor}{0pt}{}{n-1}{k}^{\alpha,\beta}$.
• The numbers $\genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta}$ can be expressed using elementary symmetric functions through generating functions and weight assignments.
• A multinomial convolution identity holds: $\binom{k}{k_1,\ldots,k_p} \genfrac{\lfloor}{\rfloor}{0pt}{}{n}{k}^{\alpha,\beta} = \sum_{l_1+\cdots+l_p=n} \binom{n}{l_1,\ldots,l_p} \genfrac{\lfloor}{\rfloor}{0pt}{}{l_1}{k_1}^{\alpha,\beta} \cdots \genfrac{\lfloor}{\rfloor}{0pt}{}{l_p}{k_p}^{\alpha,\beta}$.
• A second convolution identity is proven by analyzing the insertion of $s$ last elements into $k$ lists, yielding a sum over ordered indices $i_1 \leq \cdots \leq i_{s-j}$ with weights $\left(\alpha + \beta\right)i_l + \alpha(m - (s-j-l))$.