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Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include k -Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan functions indexed by partition weight are the characters of U q (sl ˆ ℓ) -generalized Demazure crystals as studied by Lakshmibai-Littelmann-Magyar and Naoi. We obtain Schur positive formulas for these functions, settling conjectures of Chen-Haiman and Shimozono-Weyman. Our approach more generally gives key positive formulas for graded Euler characteristics of certain vector bundles on Schubert varieties by matching them to characters of generalized Demazure crystals. [ABSTRACT FROM AUTHOR]

Partitions, the partition function p(n), and the hook lengths of their Ferrers–Young diagrams are important objects in combinatorics, number theory, and representation theory. For positive integers n and t, we study p t e (n) (resp. p t o (n) ), the number of partitions of n with an even (resp. odd) number of t-hooks. We study the limiting behavior of the ratio p t e (n) / p (n) , which also gives p t o (n) / p (n) , since p t e (n) + p t o (n) = p (n) . For even t, we show that lim n → ∞ p t e (n) p (n) = 1 2 , and for odd t, we establish the non-uniform distribution lim n → ∞ p t e (n) p (n) = 1 2 + 1 2 (t + 1) / 2 if 2 ∣ n , 1 2 - 1 2 (t + 1) / 2 otherwise. Using the Rademacher circle method, we find an exact formula for p t e (n) and p t o (n) , and this exact formula yields these distribution properties for large n. We also show that for sufficiently large n, the sign of p t e (n) - p t o (n) is periodic. [ABSTRACT FROM AUTHOR]

Nicolas [8] and DeSalvo and Pak [3] proved that the partition function p(n) is log concave for n ≥ 25 . Chen et al. [2] proved that p(n) satisfies the third order Turán inequality, and that the associated degree 3 Jensen polynomials are hyperbolic for n ≥ 95 . Recently, Griffin et al. [5] proved more generally that for all d, the degree d Jensen polynomials associated to p(n) are hyperbolic for sufficiently large n. In this paper, we prove that the same result holds for the k-regular partition function p k (n) for k ≥ 2 . In particular, for any positive integers d and k, the order d Turán inequalities hold for p k (n) for sufficiently large n. The case when d = k = 2 proves a conjecture by Neil Sloane that p 2 (n) is log concave. [ABSTRACT FROM AUTHOR]

A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular. [ABSTRACT FROM AUTHOR]