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We give a self-contained proof of the strongest version of Mason's conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave.
“Matroids: unimodal conjectures and Motzkin’s theorem”
John Mason · 1972
Earlier work this paper cites.
“Matroids, hypergraphs, and the max-flow min-cut theorem”, 1975
Paul Seymour · 1975
Earlier work this paper cites.
“Combinatorial applications of an inequality from statistical mechanics”
Paul Seymour and Dominic Welsh · 1975
Earlier work this paper cites.
“On the independent set numbers of a finite matroid”
Thomas Dowling · 1980
Earlier work this paper cites.
“On the unimodality of the independent set numbers of a class of matroids”
Carolyn Mahoney · 1985
Earlier work this paper cites.
“A conjecture on matroids”
Cui Zhao · 1985
Earlier work this paper cites.
“On the independence numbers of a matroid”
Yahya Hamidoune and Isabelle Salaün · 1989
Earlier work this paper cites.
“Towards a theory of negative dependence”
Robin Pemantle · 2000
Cited alongside, same era.
“Negative Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids”
David. Wagner · 2008
Cited alongside, same era.
“Negative dependence and the geometry of polynomials”
Julius Borcea, Petter Brändén and Thomas. Liggett · 2009
Cited alongside, same era.
“On multivariate Newton-like inequalities”
Leonid Gurvits · 2009
Cited alongside, same era.
“Negative correlation and log-concavity”
Jeff Kahn and Michael Neiman · 2010
Cited alongside, same era.
“A strong log-concavity property for measures on Boolean algebras”
Jeff Kahn and Michael Neiman · 2011
Cited alongside, same era.
“Log-concavity of characteristic polynomials and the Bergman fan of matroids”
June Huh and Eric Katz · 2012
Later among the works it cites.
“The f-vector of a representable-matroid complex is log-concave”
Matthias Lenz · 2013
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“Hodge theory for combinatorial geometries”
Karim Adiprasito, June Huh and Eric Katz · 2018
Closest in time.
“Log-concave polynomials, entropy, and a deterministic approximation algorithm for counting bases of matroids” to appear
Nima Anari, Shayan Oveis Gharan and Cynthia Vinzant · 2018
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“The geometry of matroids”
Federico Ardila · 2018
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“Correlation bounds for fields and matroids” https://arxiv.org/abs/1806.02675, 2018
June Huh, Benjamin Schröter and Botong Wang · 2018
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James Oxley · 2011
Cited alongside, same era.
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