Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.2

Combinatorial Proofs of Some Stirling Number Convolution Formulas

Mark Shattuck
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.

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(Concerned with sequences A008275 A008277 A008297.)

Received August 25 2021; revised version received January 9 2022. Published in Journal of Integer Sequences, January 29 2022.

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