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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]

We make a broad conjecture about the k -Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the k -Schur expansion of (1) Hall-Littlewood polynomials, proving the q = 0 case of the strengthened Macdonald positivity conjecture from [24] ; (2) the product of a Schur function and a k -Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum cohomology of complete flag varieties; (3) k -split polynomials, solving a substantial special case of a problem of Broer and Shimozono-Weyman on parabolic Hall-Littlewood polynomials [37]. In addition, we prove the conjecture that the k -Schur functions defined via k -split polynomials [25] agree with those defined in terms of strong tableaux [21]. [ABSTRACT FROM AUTHOR]