Strange expectations and simultaneous cores

Let gcd(a, b) = 1. J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is max λ∈core(a,b) (size(λ)) = (a 2 − 1)(b 2 − 1) 24 , and that this maximum was achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong's conjecture that the expected number of boxes in a simultaneous (a, b)-core is E λ∈core(a,b) (size(λ)) = (a − 1)(b − 1)(a + b + 1) 24. We extend P. Johnson's method to compute the variance to be V λ∈core(a,b) (size(λ)) = ab(a − 1)(b − 1)(a + b)(a + b + 1) 1440 , and also prove polynomiality of all moments. By extending the definitions of " simultaneous cores " and " number of boxes " to affine Weyl groups, we give uniform generalizations of all three formulae above to simply-laced affine types. We further explain the appearance of the number 24 using the " strange formula " of H. Freudenthal and H. de Vries.

Arxiv preprint arXiv:0807.4727, 2008

Let r j (π, s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of π mod s is defined as

Contemporary mathematics, 2018

AMS: Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Arithmetic aspects of modular and Shimura varieties. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Rational points. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Applications to coding theory and cryptography. msc | Number theory-Discontinuous groups and automorphic forms-Automorphic forms on. msc | Number theory-Algebraic number theory: global fields-Class field theory. msc | Number theory-Zeta and L-functions: analytic theory-None of the above, but in this section. msc | Algebraic geometry-Arithmetic problems. Diophantine geometry-Modular and Shimura varieties. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture. msc | Number theory-Arithmetic algebraic geometry (Diophantine geometry)-Curves of arbitrary genus or genus = 1 over global fields.

Journal d'Analyse Mathématique, 2007

We introduce two notions of complexity of a system of polynomials p 1 ,. .. , p r ∈ Z[n] and apply them to characterize the limits of the expressions of the form µ(A 0 ∩T −p 1 (n) A 1 ∩. .. ∩ T −p r (n) A r) where T is a skew-product transformation of a torus T d and A i ⊆ T d are measurable sets. The dynamical results obtained allow us to construct subsets of integers with specific combinatorial properties related to the polynomial Szemerédi theorem.

Journal of Stochastic Analysis

We describe the random variables, having finite moments of all orders, whose number operator satisfies a quadratic equation whose positionmomentum decomposition is quadratic in the differentiation operator. We will see that the orthogonal polynomials generated by these random variables are the Jacobi polynomials.

arXiv (Cornell University), 2018

The tables of the title are a first attempt to understand empirically the sizes of certain distinguished sets, introduced by Hankyung Ko, of elements in affine Weyl groups. The distinguished sets themselves each have a largest element w, and all other elements are constructible combinatorially from that largest element. The combinatorics are given in the language of right sets, in the sense of Kazhdan-Lusztig. Collectively, the elements in a given distinguished set parameterize highest weights of possible modular composition factors of the "reduction modulo p" of a p th root of unity irreducible characteristic 0 quantum group module. Here p is a prime, subject to conditions discussed below, in some cases known to be quite mild. Thus, the sizes of the distinguished sets in question are relevant to estimating how much time might be saved in any future direct approach to computing irreducible modular characters of algebraic groups from larger irreducible characters of quantum groups. Actually, Ko has described two methods for obtaining potentially effective systems of such sets. She has proved one method to work at least for all primes p as large as the Coxeter number h, in a context she indicates largely generalizes to smaller p. The other method, which produces smaller distinguished sets, is known for primes p ≥ h for which the Lusztig character formula holds, but is currently unknown to be valid without the latter condition. In the tables of this paper we calculate, for all w indexing a (p-)regular highest weight in the (p-)restricted parallelotope, distinguished set sizes for both methods, for affine types A 3 , A 4 , and A 5. To keep the printed version of this paper sufficiently small, we only use those w indexing actual restricted weights in the A 5 case. The sizes corresponding to the two methods of Ko are listed in columns (6) and (5), respectively, of the tables. We also make calculations in column (7) for a third, more "obvious" system of distinguished sets (see part (1) of Proposition 1 below), to indicate how much of an improvement each of the first two systems provides. Finally, all calculations have been recently completed for affine type A 6 , and the restricted cases listed in this paper as a final table.

Journal of stochastic analysis, 2023

For any random variable X, having finite moments of all orders, and any polynomial function f , if N denotes the number operator of X, then f (N) can be written uniquely as an infinite series of terms of the form A k (X)D k , where A k is a polynomial of degree at most k and D denotes the differentiation operator. We study the random variables for which this infinite series, called the position-momentum decomposition of X, is a finite sum meaning that after a while all the position coefficients, A k (X), vanish. Thus, f (N) belongs to the Weyl algebra of X. A simple method, for recovering the probability distribution of X, from the given finite sum position-momentum decomposition of f (N) is presented first. We apply this method to the case when f is linear and f (N) is quadratic in D, recovering the Gaussian and Gamma distributions, and their Szegö-Jacobi parameters.

Journal of Combinatorics, 2022

Proceedings of the American Mathematical Society, 2013

We prove that if u(n) denotes the number of representations of a nonnegative integer n in the form x 2 + 3y 2 with x, y ∈ Z, and a 3 (n) is the number of 3-cores of n, then u(12n + 4) = 6a 3 (n). With the help of a classical result by L. Lorenz in 1871, we also deduce that a 3 (n) = d 1,3 (3n + 1) − d 2,3 (3n + 1), where d r,3 (n) is the number of divisors of n congruent to r (mod 3), a result proved earlier by Granville and Ono by using the theory of modular forms and by Hirschhorn and Sellers with the help of elementary generating function manipulations.

Journal of Combinatorial Theory, Series A, 2012

Let ηi, i = 1,. .. , n be iid Bernoulli random variables, taking values ±1 with probability 1 2. Given a multiset V of n elements v1,. .. , vn of an additive group G, we define the concentration probability of V as ρ(V) := sup v∈G P(η1v1 +. .. ηnvn = v). An old result of Erdős and Moser asserts that if vi are distinct real numbers then ρ(V) is O(n − 3 2 log n). This bound was then refined by Sárközy and Szemerédi to O(n − 3 2), which is sharp up to a constant factor. The ultimate result dues to Stanley who used tools from algebraic geometry to give a complete description for sets having optimal concentration probability; the result now becomes classic in algebraic combinatorics. In this paper, we will prove that the optimal sets from Stanley's work are stable. More importantly, our result gives an almost complete description for sets having large concentration probability.