## Abstract

Let *L* be a finite distributive lattice and µ: *L* → ℝ^{+} a log-supermodular function. For functions *k*: *L* → ℝ^{+} let

We prove for any pair *g,h*: *L* → ℝ^{+} of monotonely increasing functions, that

where “≪” denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to *q*=1.

The polynomial FKG inequality has applications to *f*-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of Schubert varieties, and to correlation-type inequalities for a class of power series weighted by Young tableaux. This class contains series involving Plancherel measure for the symmetric groups and its poissonization.

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Research supported by the Knut and Alice Wallenberg Foundation, grant KAW.2005.0098.

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Björner, A. A *q*-analogue of the FKG inequality and some applications.
*Combinatorica* **31, **151 (2011). https://doi.org/10.1007/s00493-011-2644-1

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### Mathematics Subject Classification (2000)

- 05A20
- 05E10
- 60C05