Unimodal polynomials arising from symmetric functions

Brenti, Francesco

doi:10.1090/S0002-9939-1990-0993741-2

Unimodal polynomials arising from symmetric functions
HTML articles powered by AMS MathViewer

by Francesco Brenti PDF

Proc. Amer. Math. Soc. 108 (1990), 1133-1141 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

Determinants, paths, and plane partitions

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

Generalized

-vectors, intersection cohomology of toric varieties and related results

11

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