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On the Gap Structure of Generalized Stirling Numbers

Jianru Shen, Udita N. Katugampola

Abstract

Katugampola's 2015 study of generalized fractional differential operators produced triangular arrays of integer coefficients indexed by a fractional order r and by dimensions n and k, but no combinatorial interpretation has been established for any fractional order. We give the first such interpretation, with two main results: (i) a complete combinatorial interpretation for r = 1/2 and n = 1,2,3, and (ii) a rigorous proof that this interpretation cannot extend to n >= 4 within the same framework. For n = 1,2,3, we show that the coefficients for r = 1/2 count binary sequences satisfying two conditions: they contain at least one symbol B, and they have gap <= 1, where the gap is the distance between the first and last occurrence of B. Each sequence is assigned a type k by a parity-dependent rule involving the gap value, and exhaustive enumeration matches Katugampola's coefficients exactly. We then prove an obstruction theorem showing that the gap <= 1 condition forces any such model to produce at most two distinct types per row, whereas Katugampola's array requires at least three types for every n >= 4. Thus the gap <= 1 binary-sequence interpretation works if and only if n = 1,2,3. Our results turn a computational observation into a rigorous impossibility theorem and provide guidance for future attempts to obtain complete combinatorial interpretations of fractional-calculus coefficients.

On the Gap Structure of Generalized Stirling Numbers

Abstract

Katugampola's 2015 study of generalized fractional differential operators produced triangular arrays of integer coefficients indexed by a fractional order r and by dimensions n and k, but no combinatorial interpretation has been established for any fractional order. We give the first such interpretation, with two main results: (i) a complete combinatorial interpretation for r = 1/2 and n = 1,2,3, and (ii) a rigorous proof that this interpretation cannot extend to n >= 4 within the same framework. For n = 1,2,3, we show that the coefficients for r = 1/2 count binary sequences satisfying two conditions: they contain at least one symbol B, and they have gap <= 1, where the gap is the distance between the first and last occurrence of B. Each sequence is assigned a type k by a parity-dependent rule involving the gap value, and exhaustive enumeration matches Katugampola's coefficients exactly. We then prove an obstruction theorem showing that the gap <= 1 condition forces any such model to produce at most two distinct types per row, whereas Katugampola's array requires at least three types for every n >= 4. Thus the gap <= 1 binary-sequence interpretation works if and only if n = 1,2,3. Our results turn a computational observation into a rigorous impossibility theorem and provide guidance for future attempts to obtain complete combinatorial interpretations of fractional-calculus coefficients.
Paper Structure (17 sections, 7 theorems, 15 equations, 4 tables)

This paper contains 17 sections, 7 theorems, 15 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Binary Sequences and the Gap Function
  3. Binary Sequences
  4. The Gap Function
  5. Valid Sequences
  6. The Type Function
  7. The Cases $n=1$ and $n=2$
  8. The Case $n=1$
  9. The Case $n=2$
  10. The Case $n=3$
  11. The Success of the Gap $\leq 1$ Structure
  12. Structural Obstruction for $n \geq 4$
  13. Computational Evidence
  14. Type Count Limitation
  15. Obstruction Theorem
  16. ...and 2 more sections

Key Result

Theorem 3.1

For $n=1$ and $k=1$, we have $c_{1,1}^{(1/2)} = |\{\sigma \in V_1 : \tau_1(\sigma) = 1\}| = 1.$

Theorems & Definitions (31)

  • Definition 2.1: Binary Sequence
  • Remark 2.2
  • Example 2.3
  • Definition 2.4: Position Functions
  • Definition 2.5: Gap Function
  • Remark 2.6
  • Example 2.7
  • Definition 2.8: Valid Sequences
  • Remark 2.9
  • Example 2.10
  • ...and 21 more