Unimodal polynomials arising from symmetric functions
HTML articles powered by AMS MathViewer
- by Francesco Brenti
- Proc. Amer. Math. Soc. 108 (1990), 1133-1141
- DOI: https://doi.org/10.1090/S0002-9939-1990-0993741-2
- PDF | Request permission
Abstract:
We present a general result that, using the theory of symmetric functions, produces several new classes of symmetric unimodal polynomials. The result has applications to enumerative combinatorics including the proof of a conjecture by R. Stanley.References
- George E. Andrews, A theorem on reciprocal polynomials with applications to permutations and compositions, Amer. Math. Monthly 82 (1975), no. 8, 830–833. MR 396507, DOI 10.2307/2319803
- Francesco Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989), no. 413, viii+106. MR 963833, DOI 10.1090/memo/0413
- Francesco Brenti, Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 81 (1989), no. 413, viii+106. MR 963833, DOI 10.1090/memo/0413
- Lynne M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc. 101 (1987), no. 4, 771–775. MR 911049, DOI 10.1090/S0002-9939-1987-0911049-8
- Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
- A. M. Garsia and J. Remmel, A combinatorial interpretation of $q$-derangement and $q$-Laguerre numbers, European J. Combin. 1 (1980), no. 1, 47–59. MR 576766, DOI 10.1016/S0195-6698(80)80021-7 I. Gessel and G. Viennot, Determinants, paths, and plane partitions, preprint.
- I. P. Goulden and D. M. Jackson, Combinatorial enumeration, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1983. With a foreword by Gian-Carlo Rota. MR 702512
- I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
- Robert A. Proctor, Representations of ${\mathfrak {s}}{\mathfrak {l}}(2,\,\textbf {C})$ on posets and the Sperner property, SIAM J. Algebraic Discrete Methods 3 (1982), no. 2, 275–280. MR 655567, DOI 10.1137/0603026
- Robert A. Proctor, Solution of two difficult combinatorial problems with linear algebra, Amer. Math. Monthly 89 (1982), no. 10, 721–734. MR 683197, DOI 10.2307/2975833
- Robert A. Proctor, Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin. 5 (1984), no. 4, 331–350. MR 782055, DOI 10.1016/S0195-6698(84)80037-2
- Bruce E. Sagan, Inductive and injective proofs of log concavity results, Discrete Math. 68 (1988), no. 2-3, 281–292. MR 926131, DOI 10.1016/0012-365X(88)90120-3
- Rodica Simion, A multi-indexed Sturm sequence of polynomials and unimodality of certain combinatorial sequences, J. Combin. Theory Ser. A 36 (1984), no. 1, 15–22. MR 728500, DOI 10.1016/0097-3165(84)90075-X
- Richard P. Stanley, Theory and application of plane partitions. I, II, Studies in Appl. Math. 50 (1971), 167–188; ibid. 50 (1971), 259–279. MR 325407, DOI 10.1002/sapm1971503259
- Richard P. Stanley, Unimodal sequences arising from Lie algebras, Combinatorics, representation theory and statistical methods in groups, Lecture Notes in Pure and Appl. Math., vol. 57, Dekker, New York, 1980, pp. 127–136. MR 588199
- Richard P. Stanley, Unimodality and Lie superalgebras, Stud. Appl. Math. 72 (1985), no. 3, 263–281. MR 790132, DOI 10.1002/sapm1985723263 —, Enumerative combinatorics, vol. 1, Wadsworth and Brooks/Cole, Monterey, California 1986. —, Generalized $h$-vectors, intersection cohomology of toric varieties and related results, Advanced Studies in Pure Math. 11 (1987), 187-213.
- Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500–535. MR 1110850, DOI 10.1111/j.1749-6632.1989.tb16434.x
- Michelle L. Wachs, On $q$-derangement numbers, Proc. Amer. Math. Soc. 106 (1989), no. 1, 273–278. MR 937015, DOI 10.1090/S0002-9939-1989-0937015-6
- David G. Wagner, The partition polynomial of a finite set system, J. Combin. Theory Ser. A 56 (1991), no. 1, 138–159. MR 1082848, DOI 10.1016/0097-3165(91)90027-E
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 1133-1141
- MSC: Primary 05A15; Secondary 05A05, 05A30
- DOI: https://doi.org/10.1090/S0002-9939-1990-0993741-2
- MathSciNet review: 993741