### Abstract

The Legendre-Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical secondorder Legendre differential expression. Specifically, these numbers are the coef-ficients of integral composite powers of the Legendre expression in Lagrangian symmetric form. Quite remarkably, they share many similar properties with the classical Stirling numbers of the second kind which, as shown by Littlejohn and Wellman, are the coefficients of integral powers of the Laguerre differential expression. An open question regarding the Legendre-Stirling numbers has been to obtain a combinatorial interpretation of these numbers. In this paper, we provide such an interpretation.

Original language | English (US) |
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Pages (from-to) | 2581-2590 |

Number of pages | 10 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 8 |

DOIs | |

State | Published - Aug 1 2009 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Proceedings of the American Mathematical Society*,

*137*(8), 2581-2590. https://doi.org/10.1090/S0002-9939-09-09814-1