On the total positivity of restricted Stirling numbers

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Abstract

This note shows that the matrix whose (n,k) entry is the number of set partitions of {1,,n} into k blocks with size at most m is never totally positive for m3; thus answering a question posed in [H. Han, S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, European J. Combin. 29 (2008) 1544–1554].

Highlights

► We study totally positive properties of restricted Stirling numbers. ► The matrix of restricted Stirling number is totally positive for m2 and m=. ► The matrix of restricted Stirling number is not totally positive for m3.

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This paper is part of the author’s Ph.D. Thesis written under the direction of Prof. F. Brenti at the Univ. “la Sapienza” of Rome, Italy.