Related papers: On some monotonic combinatorial sequences conjectu…

In this note we present a collection of attempts of some researchers to prove or disprove whether the Zeraoulia sequences convergent. Even nowdays convergence of Zeraoulia sequences still open.

Using a new technique, we prove a rich family of special cases of the matroid intersection conjecture. Roughly, we prove the conjecture for pairs of tame matroids which have a common decomposition by 2-separations into finite parts.

We will give a pure combinatorial proof of the Eisenbud-Goto conjecture for arbitrary monomial curves. Moreover, we will show that the conjecture holds for certain simplicial affine semigroup rings.

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

In this paper, we study some supercongruences involving the sequence $$ t_n(x)=\sum_{k=0}^n\binom{n}{k}\binom{x}{k}\binom{x+k}{k}2^k $$ and solve some open problems. For any odd prime $p$ and $p$-adic integer $x$, we determine…

We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.

Harmonic numbers are significant in various branches of number theory. With the help of the digamma function, we prove ten conjectural series of Z.-W. Sun involving harmonic numbers. Several ones of them are also series expansions of…

Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.

Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.

We provide a simple proof for the union-closed sets conjecture, a long-standing open problem in set theory with immediate applications to graph theory, number theory, and order-theory.

In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive…

In his study of Ramanujan-Sato type series for $1/\pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials…

We make a summary of the different types of proofs adding some new ideas. In addition we conjecture some relations which could be necessary in "modular type proofs" (not still found) of the Ramanujan-like series for 1/\pi^2.

We obtain new partial results supporting the spectral set conjecture in dimension 1.

Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We…

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some…

We establish a new simple explicit description of combinatorial wall-crossing for the rational Cherednik algebra applied to the trivial representation. In this way we recover a theorem of P. Dimakis and G. Yue. We also present two…

Let $p$ be a prime and $c,d\in\mathbb{Z}$. Sun introduced the determinant $D_p^-(c,d):=\det[(i^2+cij+dj^2)^{p-2}]_{1<i,j<p-1}$ for $p>3$. In this paper, we confirm three conjectures on $D_p^-(c,d)$ proposed by Zhi-Wei Sun.

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of…