publication . Article . Preprint . 2012

Jacobi-Stirling polynomials and $P$-partitions

Gessel, Ira M.; Lin, Zhicong; Zeng, Jiang;
Open Access English
  • Published: 03 Jan 2012
  • Publisher: Elsevier
Abstract
18 pages, 4 figures, 1 table, minor modifications, to appear in European Journal of Combinatorics, 2012; International audience; We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind $ \JS(n+k,n;z)$ by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting $\JS(n+k,n;z)=p_{k,0}(n)+p_{k,1}(n)z+...+p_{k,k}(n)z^k$, we show that $(1-t)^{3k-i+1}\sum_{n\geq0}p_{k,i}(n)t^n$ is a polynomial in $t$ with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley's theory of $P$-partitions.
Subjects
free text keywords: 05A05, 05A15, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics - Combinatorics
16 references, page 1 of 2

[1] G. E. Andrews, E. S. Egge, W. Gawronski, L. L. Littlejohn, The Jacobi-Stirling numbers, arXiv:1112.6111.

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

[3] F. Brenti, Unimodal, Log-concave, and Po´lya Frequency Sequences in Combinatorics, Memoirs Amer. Math. Soc., No. 413, 1989.

[4] L. Comtet, Advanced Combinatorics, Boston, Dordrecht, 1974.

[5] E. S. Egge, Legendre-Stirling permutations, European J. Combin., 31 (2010) 1735-1750.

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

[7] 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. Comput. Appl. Math., 208 (2007) 29-56. [OpenAIRE]

[8] Y. Gelineau, E´tudes combinatoires des nombres de Jacobi-Stirling et d'Entringer, Th`ese de doctorat, Universit´e Claude Bernard Lyon 1, 2010.

[9] Y. Gelineau and J. Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, Electron. J. Combin., 17 (2010) R70.

[10] I. Gessel and R. P. Stanley, Stirling Polynomials, J. Combin. Theory Ser. A, 24 (1978) 24-33.

[11] J. Haglund and M. Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations, European J. Combin. 33 (2012), no. 4, 477-487. [OpenAIRE]

[12] S. Janson, M. Kuba, A. Panholzer, Generalized Stirling permutations, families of increasing trees and urn models, J. Combin. Theory Ser. A, 118 (2011) 94114.

[13] S. Park, P -Partitions and q-Stirling Numbers, J. Combin. Theory Ser. A, 68 (1994) 33-52.

[14] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., 1968.

[15] R. P. Stanley, Ordered Structures and Partitions, Mem. Amer. Math. Soc., No. 119. American Mathematical Society, Providence, R.I., 1972.

16 references, page 1 of 2
Abstract
18 pages, 4 figures, 1 table, minor modifications, to appear in European Journal of Combinatorics, 2012; International audience; We investigate the diagonal generating function of the Jacobi-Stirling numbers of the second kind $ \JS(n+k,n;z)$ by generalizing the analogous results for the Stirling and Legendre-Stirling numbers. More precisely, letting $\JS(n+k,n;z)=p_{k,0}(n)+p_{k,1}(n)z+...+p_{k,k}(n)z^k$, we show that $(1-t)^{3k-i+1}\sum_{n\geq0}p_{k,i}(n)t^n$ is a polynomial in $t$ with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley's theory of $P$-partitions.
Subjects
free text keywords: 05A05, 05A15, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics - Combinatorics
16 references, page 1 of 2

[1] G. E. Andrews, E. S. Egge, W. Gawronski, L. L. Littlejohn, The Jacobi-Stirling numbers, arXiv:1112.6111.

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

[3] F. Brenti, Unimodal, Log-concave, and Po´lya Frequency Sequences in Combinatorics, Memoirs Amer. Math. Soc., No. 413, 1989.

[4] L. Comtet, Advanced Combinatorics, Boston, Dordrecht, 1974.

[5] E. S. Egge, Legendre-Stirling permutations, European J. Combin., 31 (2010) 1735-1750.

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

[7] 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. Comput. Appl. Math., 208 (2007) 29-56. [OpenAIRE]

[8] Y. Gelineau, E´tudes combinatoires des nombres de Jacobi-Stirling et d'Entringer, Th`ese de doctorat, Universit´e Claude Bernard Lyon 1, 2010.

[9] Y. Gelineau and J. Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, Electron. J. Combin., 17 (2010) R70.

[10] I. Gessel and R. P. Stanley, Stirling Polynomials, J. Combin. Theory Ser. A, 24 (1978) 24-33.

[11] J. Haglund and M. Visontai, Stable multivariate Eulerian polynomials and generalized Stirling permutations, European J. Combin. 33 (2012), no. 4, 477-487. [OpenAIRE]

[12] S. Janson, M. Kuba, A. Panholzer, Generalized Stirling permutations, families of increasing trees and urn models, J. Combin. Theory Ser. A, 118 (2011) 94114.

[13] S. Park, P -Partitions and q-Stirling Numbers, J. Combin. Theory Ser. A, 68 (1994) 33-52.

[14] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., 1968.

[15] R. P. Stanley, Ordered Structures and Partitions, Mem. Amer. Math. Soc., No. 119. American Mathematical Society, Providence, R.I., 1972.

16 references, page 1 of 2
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Article . Preprint . 2012

Jacobi-Stirling polynomials and $P$-partitions

Gessel, Ira M.; Lin, Zhicong; Zeng, Jiang;