Inductive proofs of \(q\)-log concavity

DOI10.1016/0012-365X(92)90377-RzbMath0764.05096MaRDI QIDQ1191957

Bruce E. Sagan

Publication date: 27 September 1992

Published in: Discrete Mathematics (Search for Journal in Brave)

zbMATH Keywords

symmetric functionsGaussian polynomialsunimodality\(q\)-Stirling numbers\(q\)-concave polynomialsbibasic

Mathematics Subject Classification ID

Bell and Stirling numbers (11B73) (q)-calculus and related topics (05A30) Symmetric functions and generalizations (05E05)

Related Items (35)

The \(q\)-log-concavity and unimodality of \(q\)-Kaplansky numbers ⋮ Log-concave sequences of bi\(^{\text{s}}\)nomial coefficients with their analogs and symmetric functions ⋮ \(\sigma\)-restricted growth functions and \(p,q\)-Stirling numbers ⋮ The \(q\)-log-convexity of the Narayana polynomials of type \(B\) ⋮ The q-log-concavity of q-binomial coefficients ⋮ Some positivities in certain triangular arrays ⋮ Log-concavity and LC-positivity ⋮ Schur positivity and the \(q\)-log-convexity of the Narayana polynomials ⋮ A \(q\)-analog of approximation inclusion-exclusion ⋮ Log-convexity and strong \(q\)-\(\log\)-convexity for some triangular arrays ⋮ A planar network proof for Hankel total positivity of type \(B\) Narayana polynomials ⋮ Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials ⋮ Linear transformations preserving the strong \(q\)-log-convexity of polynomials ⋮ Effective Poset Inequalities ⋮ A local injective proof of log-concavity for increasing spanning forests ⋮ Overpartition analogues of \(q\text{-bi}^s\)nomial coefficients: basic properties and log-concavity ⋮ On the concavity properties of certain arithmetic sequences and polynomials ⋮ An analogue of Mahonian numbers and log-concavity ⋮ \(Q\)-total positivity and strong \(q\)-log-convexity for some generalized triangular arrays ⋮ \(q\)-log-concavity and \(q\)-unimodality of Gaussian polynomials and a problem of Andrews and Newman ⋮ Note on some restricted Stirling numbers of the second kind ⋮ On the maximum of the weighted binomial sum \((1 + a)^{- r} \sum_{i = 0}^r \binom{m}{i} a^i\) ⋮ Reduced matrices and q-log-concavity properties of q-Stirling numbers ⋮ Preserving log-concavity for p,q-binomial coefficient ⋮ On the log-convexity of combinatorial sequences ⋮ Log Concave Sequences of Symmetric Functions and Analogs of the Jacobi-Trudi Determinants ⋮ Preservation of log-concavity on summation ⋮ Unnamed Item ⋮ On \(q\)-analogs of descent and peak polynomials ⋮ On the q-log-concavity of Gaussian binomial coefficients ⋮ Asymptotics of the number of geodesics in the discrete Heisenberg group ⋮ Log-concavity and the exponential formula ⋮ Log-concavity results for a biparametric and an elliptic extension of the q-binomial coefficients ⋮ On Stanley's chromatic symmetric function and clawfree graphs ⋮ A maj statistic for set partitions

Cites Work

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• \(p,q\)-Stirling numbers and set partition statistics
• Reduced matrices and q-log-concavity properties of q-Stirling numbers
• Flagged Schur functions, Schubert polynomials, and symmetrizing operators
• Binomial determinants, paths, and hook length formulae
• Inductive and injective proofs of log concavity results
• A unified setting for sequencing, ranking, and selection algorithms for combinatorial objects
• The q-Stirling numbers of first and second kinds
• The q-log-concavity of q-binomial coefficients
• \(q\)-Bernoulli numbers and polynomials
• A Unimodality Result in the Enumeration of Subgroups of a Finite Abelian Group
• A Unified Setting for Selection Algorithms (II)
• Log Concave Sequences of Symmetric Functions and Analogs of the Jacobi-Trudi Determinants

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