publication . Preprint . 2009

Combinatorial interpretations of the Jacobi-Stirling numbers

Gelineau, Yoann; Zeng, Jiang;
Open Access English
  • Published: 18 May 2009
Abstract
The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.
Subjects
arxiv: Computer Science::Symbolic ComputationMathematics::History and OverviewMathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, 05A05, 05A15, 33C45 (Primary), Mathematics - Classical Analysis and ODEs, 05A10, 05A18, 34B24 (Secondary)
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1. G. E. Andrews, L. L. Littlejohn, A combinatorial interpretation of the Legendre-Stirling numbers, Proc. Amer. Math. Soc. 137 (2009), 2581-2590.

2. L. Comtet, Advanced combinatorics, Boston, Dordrecht, 1974.

3. D. Dumont, Interpr´etations combinatoires des nombres de Genocchi, , Duke math. J., t. 41, 1974, p. 305-318.

4. W. N. Everitt, L. L.Littlejohn, R. Wellman, Legendre polynomials, Legendre-Stirling numbers, and the left-definite spectral abalysis of the Legendre differential expression, J.Combut.Appl.Math., 148 (2002), 213-238. [OpenAIRE]

5. W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, G. J. Yoon, Jacobi-Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression, J.Combut.Appl.Math., 208 (2007), 29-56. [OpenAIRE]

6. Dominique Foata, Guo-Niu Han, Principes de combinatoire classique, Lecture notes, Strasbourg, 2000, revised 2008.

7. D. Foata, M. P. Schu¨tzenberger, Th´eorie g´eom´etrique des polynˆomes eul´eriens, Lecture Notes in Math no. 138, Springer-Verlag, Berlin, 1970.

8. J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., 1968.

9. R. P. Stanley, Enumerative Combinatorics, vol.2, Cambridge Studies in Advanced Mathematics, 62, 1999. Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France E-mail address: gelineau@math.univ-lyon1.fr URL: http://math.univ-lyon1.fr/~gelineau/ Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France E-mail address: zeng@math.univ-lyon1.fr URL: http://math.univ-lyon1.fr/~zeng/

Abstract
The Jacobi-Stirling numbers of the first and second kinds were introduced in 2006 in the spectral theory and are polynomial refinements of the Legendre-Stirling numbers. Andrews and Littlejohn have recently given a combinatorial interpretation for the second kind of the latter numbers. Noticing that these numbers are very similar to the classical central factorial numbers, we give combinatorial interpretations for the Jacobi-Stirling numbers of both kinds, which provide a unified treatment of the combinatorial theories for the two previous sequences and also for the Stirling numbers of both kinds.
Subjects
arxiv: Computer Science::Symbolic ComputationMathematics::History and OverviewMathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, 05A05, 05A15, 33C45 (Primary), Mathematics - Classical Analysis and ODEs, 05A10, 05A18, 34B24 (Secondary)
Download from

1. G. E. Andrews, L. L. Littlejohn, A combinatorial interpretation of the Legendre-Stirling numbers, Proc. Amer. Math. Soc. 137 (2009), 2581-2590.

2. L. Comtet, Advanced combinatorics, Boston, Dordrecht, 1974.

3. D. Dumont, Interpr´etations combinatoires des nombres de Genocchi, , Duke math. J., t. 41, 1974, p. 305-318.

4. W. N. Everitt, L. L.Littlejohn, R. Wellman, Legendre polynomials, Legendre-Stirling numbers, and the left-definite spectral abalysis of the Legendre differential expression, J.Combut.Appl.Math., 148 (2002), 213-238. [OpenAIRE]

5. W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, G. J. Yoon, Jacobi-Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression, J.Combut.Appl.Math., 208 (2007), 29-56. [OpenAIRE]

6. Dominique Foata, Guo-Niu Han, Principes de combinatoire classique, Lecture notes, Strasbourg, 2000, revised 2008.

7. D. Foata, M. P. Schu¨tzenberger, Th´eorie g´eom´etrique des polynˆomes eul´eriens, Lecture Notes in Math no. 138, Springer-Verlag, Berlin, 1970.

8. J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., 1968.

9. R. P. Stanley, Enumerative Combinatorics, vol.2, Cambridge Studies in Advanced Mathematics, 62, 1999. Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France E-mail address: gelineau@math.univ-lyon1.fr URL: http://math.univ-lyon1.fr/~gelineau/ Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France E-mail address: zeng@math.univ-lyon1.fr URL: http://math.univ-lyon1.fr/~zeng/

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