publication . Preprint . 2011

Combinatorial interpretations of particular evaluations of complete and elementary symmetric functions

Mongelli, Pietro;
Open Access English
  • Published: 11 Nov 2011
Abstract
The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.
Subjects
arxiv: Mathematics::History and OverviewComputer Science::Symbolic ComputationMathematics::Number TheoryMathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, 05A19 (Primary) 05A05, 05A30, 11P81 (Secondary)
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[1] Adrews G. E., Gawronski W., Littlejohn L. L. (2010) The Legendre-Stirling numbers Preprint.

[2] Andrews G. E., Littlejohn L. L. (2009) A combinatorial interpretation of the Legendre-Stirling numbers. Proc. AMS, vol. 137-8 pp. 2581-2590.

[3] Brenti F. (1995) Combinatorics and Total Positivity J. Combinat. Th., series A, pp. 175-218. [OpenAIRE]

[4] Carlitz L. On abelian fields 122-136.

Trans. Amer. Math. Soc., 35 (1933), pp.

[5] Egge E. S. (2010) Legendre-Stirling permutations Europ. J. of Comb. 31, pp. 1735-1750. [OpenAIRE]

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

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

[8] Gelineau Y., Zeng J. (2010) Combinatorial interpretations of the JacobiStirling numbers. Elect. Journ. Comb. 17, #R70.

[9] Gessel I., Viennot G. Determinants, Paths and Plane Partitions. preprint (1989).

[10] Gould H. W. The q-Stirling numbers f the first and second kinds. Math. J. 28, (1961), pp. 281-289. [OpenAIRE]

Abstract
The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.
Subjects
arxiv: Mathematics::History and OverviewComputer Science::Symbolic ComputationMathematics::Number TheoryMathematics::Combinatorics
free text keywords: Mathematics - Combinatorics, 05A19 (Primary) 05A05, 05A30, 11P81 (Secondary)
Download from

[1] Adrews G. E., Gawronski W., Littlejohn L. L. (2010) The Legendre-Stirling numbers Preprint.

[2] Andrews G. E., Littlejohn L. L. (2009) A combinatorial interpretation of the Legendre-Stirling numbers. Proc. AMS, vol. 137-8 pp. 2581-2590.

[3] Brenti F. (1995) Combinatorics and Total Positivity J. Combinat. Th., series A, pp. 175-218. [OpenAIRE]

[4] Carlitz L. On abelian fields 122-136.

Trans. Amer. Math. Soc., 35 (1933), pp.

[5] Egge E. S. (2010) Legendre-Stirling permutations Europ. J. of Comb. 31, pp. 1735-1750. [OpenAIRE]

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

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

[8] Gelineau Y., Zeng J. (2010) Combinatorial interpretations of the JacobiStirling numbers. Elect. Journ. Comb. 17, #R70.

[9] Gessel I., Viennot G. Determinants, Paths and Plane Partitions. preprint (1989).

[10] Gould H. W. The q-Stirling numbers f the first and second kinds. Math. J. 28, (1961), pp. 281-289. [OpenAIRE]

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