Related papers: Kirillov's unimodality conjecture for the rectangu…

We establish the unimodality and the asymptotic strong unimodality of the ordinary multinomials and give their smallest mode leading to the expression of the maximal probability of convolution powers of the discrete uniform distribution. We…

We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot $4_1$ in arbitrary rectangular representation $R=[r^s]$ as a sum over all Young sub-diagrams $\lambda$ of $R$ with extraordinary simple…

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials ${{g}_{n}}\left( x \right)={{\left( 1-x…

The conjecture by Steklov was solved negatively by Rakhmanov in 1979. His original proof was based on the formula for orthogonal polynomial obtained by adding point masses to the measure of orthogonality. In this note, we show how this…

We obtain the Kirillov vector fields on the set of functions $f$ univalent inside the unit disk, in terms of the Faber polynomials of $1/f(1/z)$. Our construction relies on the generating function for Faber polynomials.

The descent polynomials of separable permutations and derangements are both demonstrated to be unimodal. Moreover, we prove that the $\gamma$-coefficients of the first are positive with an interpretation parallel to the classical Eulerian…

It was conjectured in arXiv:1211.2788 that bosonization of singular vectors (in Neveu-Schwarz sector) of $\mathcal{N}=1$ super analog of the Virasoro algebra can be identified with Uglov symmetric function. In the paper we prove this…

We prove a conjecture of Bourn and Willenbring (2020) regarding the palindromicity and unimodality of a certain family of polynomials $N_n(t)$. These recursively defined polynomials arise as the numerators of generating functions in the…

I. P. Goulden, S. Litsyn, and V. Shevelev [On a sequence arising in algebraic geometry, J. Integer Sequences 8 (2005), 05.4.7] conjectured that certain Laurent polynomials associated with the solution of a functional equation have only odd…

We show that with high probability the number of real zeroes of a random polynomial is bounded by the number of vertices on its Newton-Hadamard polygon times the cube of the logarithm of the polynomial degree. A similar estimate holds for…

The Eulerian transformation is the linear operator on polynomials in one variable with real coefficients which maps the powers of this variable to the corresponding Eulerian polynomials. The derangement transformation is defined similarly.…

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both…

We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture and previously studied by Panova and the…

Canon permutations are permutations of the multiset having $k$ copies of each integer between $1$ and $n$, with the property that the subsequences obtained by taking the $j$th copy of each entry, for each fixed $j$, are all the same. For…

The goal of this paper is to attract attention of the reader to a dimension-free geometric inequality that can be proved using the classical needle decomposition. This inequality allows us to derive sharp dimension-free estimates for the…

We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima…

In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients.…

Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective…

We show that certain Riordan arrays have generating functions that can be expressed as continued fractions of Jacobi and Thron type. We investigate the inverses of such arrays, which in certain circumstances can also have generating…

We give a direct proof for the positivity of Kirillov's character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group $G$. Then we use this positivity result to construct…