2019/02/14 by Pinelis, Iosif · 1 citation
#05B35 #06A07 #26D15 #26D20 #60C05 #60E15 #62H05 #62H10 #82D99 #90C27 #Combinatorics (math.CO) #FOS: Mathematics #Primary 06D99 #Probability (math.PR) #Statistics Theory (math.ST) #secondary 05A20
paper · doi:10.48550/arxiv.1902.05520
A notion of generalized n-semimodularity is introduced, which extends that of (sub/super)mod\-ularity in four ways at once. The main result of this paper, stating that every generalized (n\colon 2)-semimodular function on the nth Cartesian power of a distributive lattice is generalized n-semimodular, may be considered a multi/infinite-dimensional analogue of the well-known Muirhead lemma in the theory of Schur majorization. This result is also similar to a discretized version of the well-known theorem due to Lorentz, which latter was given only for additive-type functions. Illustrations of our main result are presented for counts of combinations of faces of a polytope; one-sided potentials; multiadditive forms, including multilinear ones -- in particular, permanents of rectangular matrices and elementary symmetric functions; and association inequalities for order statistics. Based on an extension of the FKG inequality due to Rinott & Saks and Aharoni & Keich, applications to correlation inequalities for order statistics are given as well.