On the Gap Structure of Generalized Stirling Numbers

TL;DR

This paper provides the first combinatorial interpretation for a special case of generalized Stirling numbers related to fractional calculus and proves that such an interpretation cannot extend beyond small dimensions, guiding future research.

Contribution

It introduces a combinatorial interpretation for r=1/2 and small n, and proves an impossibility result for extending this interpretation to larger n.

Findings

Binary sequences with gap <= 1 count coefficients for n=1,2,3.

The interpretation cannot extend to n >= 4 due to a type limitation.

The results turn a computational pattern into a rigorous impossibility theorem.

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.