Combinatorial Proofs of Some Stirling Number Convolution Formulas
Department of Mathematics
University of Tennessee
Knoxville, TN 37996-1320
Recently, some new convolution formulas extending the orthogonality of
the Stirling numbers of the first and second kind were shown by algebraic
techniques. The formulas involve sums of products of the two Stirling
numbers wherein the inner arguments vary while differing by a prescribed
amount and the outer arguments are fixed. Here, we provide combinatorial
proofs of these formulas using direct enumeration and sign-changing
involutions. Our arguments may be extended to establish generalizations
of the foregoing results in terms of the r-Stirling numbers.
Full version: pdf,
(Concerned with sequences
Received August 25 2021; revised version received January 9 2022.
Published in Journal of Integer Sequences,
January 29 2022.
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