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On the Uniqueness Conjecture for the Maximum Stirling Numbers of the Second Kind

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Abstract

The Stirling numbers of the second kind S(n, k) satisfy

$$\begin{aligned} S(n,0)<\cdots <S(n,k_n)\ge S(n,k_n+1)>\cdots >S(n,n). \end{aligned}$$

A long standing conjecture asserts that there exists no \(n\ge 3\) such that \(S(n,k_n)=S(n,k_n+1)\). In this note, we give a characterization of this conjecture in terms of multinomial probabilities, as well as sufficient conditions on n ensuring that \(S(n,k_n)>S(n,k_n+1)\).

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Authors and Affiliations

  1. Departamento de Métodos Estadísticos, Universidad de Zaragoza, 50009, Zaragoza, Spain

    José A. Adell

  2. Departamento de Matemáticas, Universidad de Jaén, 23071, Jaén, Spain

    Daniel Cárdenas-Morales

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  1. José A. Adell
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  2. Daniel Cárdenas-Morales
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Correspondence to Daniel Cárdenas-Morales.

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This work is partially supported by Research Project PGC2018-097621-B-I00. The second author is also supported by Junta de Andalucía Research Group FQM-0178.

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Adell, J.A., Cárdenas-Morales, D. On the Uniqueness Conjecture for the Maximum Stirling Numbers of the Second Kind. Results Math 76, 93 (2021). https://doi.org/10.1007/s00025-021-01393-7

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