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A convolution for complete and elementary symmetric functions. (English) Zbl 1278.05242

Summary: In this paper we give a convolution identity for complete and elementary symmetric functions. This result can be used to prove and discover some combinatorial identities involving \(r\)-Stirling numbers, \(r\)-Whitney numbers and \(q\)-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde’s convolution identity.

MSC:

05E05 Symmetric functions and generalizations
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
11B73 Bell and Stirling numbers
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References:

[1] Andrews G.E.: The Theory of Partitions. Addison-Wesley Publishing, Boston (1976) · Zbl 0371.10001
[2] Benoumhani M.: On Whitney numbers of Dowling lattices. Disc. Math. 159, 13-33 (1996) · Zbl 0861.05004 · doi:10.1016/0012-365X(95)00095-E
[3] Benoumhani M.: On some numbers related to Whitney numbers of Dowling lattices. Adv. Appl. Math. 19, 106-116 (1997) · Zbl 0876.05001 · doi:10.1006/aama.1997.0529
[4] Benoumhani M.: Log-concavity of Whitney numbers of Dowling lattices. Adv. Appl. Math. 22, 186-189 (1999) · Zbl 0918.05003 · doi:10.1006/aama.1998.0621
[5] Broder A.Z.: The r-Stirling numbers. Disc. Math. 49, 241-259 (1984) · Zbl 0535.05006 · doi:10.1016/0012-365X(84)90161-4
[6] Cheon G.-S., Jung J.-H.: r-Whitney numbers of Dowling lattices. Disc. Math. 312, 2337-2348 (2012) · Zbl 1246.05009 · doi:10.1016/j.disc.2012.04.001
[7] Dowling T.A.: A class of geometric lattices based on finite groups. J. Combin. Theory Ser. B 14, 61-86 (1973) · Zbl 0247.05019 · doi:10.1016/S0095-8956(73)80007-3
[8] Konvalina J.: A generalization of Waring’s formula. J. Combin. Theory Ser. A. 75(2), 281-294 (1996) · Zbl 0857.05094 · doi:10.1006/jcta.1996.0078
[9] Kuba M., Prodinger H.: A note on Stirling series. Integers 10, 393-406 (2010) · Zbl 1268.11036 · doi:10.1515/integ.2010.034
[10] Macdonald I.G.: Symmetric Functions and Hall Polynomials. Clarendon Press, Oxford (1995) · Zbl 0824.05059
[11] Merca, M.: A special case of the generalized Girard-Waring formula. J. Integer Seq. 15 (2012). Article 12.5.7 · Zbl 1292.05261
[12] Mező I.: On the maximum of r-Stirling numbers. Adv. Appl. Math. 41(3), 293-306 (2008) · Zbl 1165.11023 · doi:10.1016/j.aam.2007.11.002
[13] Mező I.: New properties of r-Stirling series. Acta Math. Hungar. 119, 341-358 (2008) · Zbl 1174.11026 · doi:10.1007/s10474-007-7047-9
[14] Mező I.: A new formula for the Bernoulli polynomials. Results Math. 58, 329-335 (2010) · Zbl 1237.11010 · doi:10.1007/s00025-010-0039-z
[15] Stanley R.P.: Enumerative Combinatorics 1. Cambridge University Press, Cambridge (1997) · Zbl 0889.05001 · doi:10.1017/CBO9780511805967
[16] Zeng J.: On a generalization of Warings formula. Adv. Appl. Math. 19, 450-452 (1997) · Zbl 0899.05069 · doi:10.1006/aama.1997.0545
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