Schur positivity and the q-log-convexity of the Narayana polynomials
William Y.C. Chen
, Larry X.W. Wang
and Arthur L.B. Yang
DOI: 10.1007/s10801-010-0216-x
Abstract
We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers N q ( n, k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.
Pages: 303–338
Keywords: keywords $q$-log-concavity; $q$-log-convexity; $q$-narayana number; narayana polynomial; lattice permutation; Schur positivity; Littlewood-Richardson rule
Full Text: PDF
References
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2. Bergeron, F., Biagioli, R., Rosas, M.: Inequalities between Littlewood-Richardson coefficients. J. Comb. Theory Ser. A 113(4), 567-590 (2006)
3. Brändén, P.: q-Narayana numbers and the flag h-vector of J (2 \times n). Discrete Math. 281, 67-81 (2004)
4. Brenti, F.: Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Am. Math. Soc. 81(413), 1-106 (1989)
5. Brenti, F.: Log-concave and unimodal sequences in algebra, combinatorics and geometry: an update. Contemp. Math. 178, 71-89 (1994)
6. Butler, L.M.: The q-log-concavity of q-binomial coefficients. J. Comb. Theory Ser. A 54, 54-63 (1990)
7. Butler, L.M., Flanigan, W.P.: A note on log-convexity of q-Catalan numbers. Ann. Comb. 11, 369-373 (2007)
8. Carlitz, L., Riordan, J.: Two element lattice permutations and their q-generalization. Duke Math. J. 31, 371-388 (1964)
9. Davenport, H., Pólya, G.: On the product of two power series. Can. J. Math. 1, 1-5 (1949)
10. Deutsch, E.: A bijection on Dyck paths and its consequences. Discrete Math. 179, 253-256 (1998)
11. Fomin, S., Fulton, W., Li, C.-K., Poon, Y.-T.: Eigenvalues, singular values, and Littlewood- Richardson coefficients. Am. J. Math. 127, 101-127 (2005)
12. Fürlinger, J., Hofbauer, J.: q-Catalan numbers. J. Comb. Theory Ser. A 40, 248-264 (1985)
13. Kirillov, A.N.: Completeness of states of the generalized Heisenberg magnet. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. im Steklova (LOMI) 134, 169-189 (1984)
14. Kleber, M.: Plücker relations on Schur functions. J. Algebraic Comb. 13, 199-211 (2001)
15. Kostov, V.P., Martínez-Finkelshtein, A., Shapiro, B.Z.: Narayana numbers and Schur-Szegö composition. J. Approx. Theory 161, 464-476 (2009) J Algebr Comb (2010) 32: 303-338
16. Krattenthaler, C.: On the q-log-concavity of Gaussian binomial coefficients. Monatsh. Math. 107, 333-339 (1989)
17. Lam, T., Pylyavaskyy, P.: Cell transfer and monomial positivity. J. Algebraic Comb. 26, 209-224 (2007)
18. Lam, T., Postnikov, A., Pylyavskyy, P.: Schur positivity and Schur log-concavity. Am. J. Math. 129, 1611-1622 (2007)
19. Liu, L., Wang, Y.: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39, 453-476 (2007)
20. Leroux, P.: Reduced matrices and q-log-concavity properties of q-Stirling numbers. J. Comb. Theory Ser. A 54, 64-84 (1990)
21. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
22. McNamara, P.R.W., Sagan, B.E.: Infinite log-concavity: developments and conjectures. Adv. Appl. Math. 44, 1-15 (2010)
23. Okounkov, A.: Log-concavity of multiplicities with applications to characters of U(\infty ). Adv. Math. 127, 258-282 (1997)
24. Remmel, J.B., Whitehead, T.: On the Kronecker product of Schur functions of two row shapes. Bull. Belg. Math. Soc. Simon Stevin 1, 649-683 (1994)
25. Rosas, M.H.: The Kronecker product of Schur functions indexed by two-row shapes or hook shapes. J. Algebraic Comb. 14, 153-173 (2001)
26. Sagan, B.E.: Inductive proofs of q-log concavity. Discrete Math. 99, 298-306 (1992)
27. Sagan, B.E.: Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants. Trans. Am. Math. Soc. 329, 795-811 (1992)
28. Stanley, R.P.: Theory and applications of plane partitions, part
2. Stud. Appl. Math. 50, 259-279 (1971)
29. Stanley, R.P.: Log-concave and unimodal sequences in algebra, combinatorics and geometry. Ann. N.Y. Acad. Sci. 576, 500-535 (1989)
30. Stanley, R.P.: Enumerative Combinatorics, vol.
2. Cambridge University Press, Cambridge (1999)
31. Stembridge, J.R.: Multiplicity-free products of Schur functions. Ann. Comb. 5, 113-121 (2001)
32. Stembridge, J.R.: Counterexamples to the poset conjectures of Neggers, Stanley, and Stembridge.
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