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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]

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]

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]

We identify certain combinatorially defined rational functions which, under the shuffle to Schiffmann algebra isomorphism, map to LLT polynomials in any of the distinguished copies Λ ⁢ (X m , n) ⊂ E of the algebra of symmetric functions embedded in the elliptic Hall algebra ℰ of Burban and Schiffmann. As a corollary, we deduce an explicit raising operator formula for the ∇ operator applied to any LLT polynomial. In particular, we obtain a formula for ∇ m s λ which serves as a starting point for our proof of the Loehr–Warrington conjecture in a companion paper to this one. [ABSTRACT FROM AUTHOR]

We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for ΔhlΔ′eken, where Δ′ek and Δhl are Macdonald eigenoperators and en is an elementary symmetric function. We actually prove a stronger identity of infinite series of GLm characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions. [ABSTRACT FROM AUTHOR]

We generalize the shuffle theorem and its (km,kn) version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the (km,kn) Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of GLl characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall-Littlewood polynomials. [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]