arXiv subjects

Jovan Mikić

Publications and source records attributed to Jovan Mikić.

6 recordsLinked to original sources

arXiv2025

On a New Congruence in the Catalan Triangle

For $0\leq k \leq n$, the number $C(n,k)$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(n,k)$, using steps $(1,0)$ and $(0,1)$, that never rise above the main diagonal $y=x$. The Fuss-Catalan number of order three $C^{(3)}_n$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(2n,n)$, using steps $(1,0)$ and $(0,1)$, that do not rise above the line $y=\frac{x}{2}$. We present a new alternating convolution formula for the numbers $C(2n,k)$. By using a new class of binomial sums that we call $M$ sums, we prove that this sum is divisible by $C^{(3)}_n$ and by the central binomial coefficient $\binom{2n}{n}$. We do this by examining the numbers $T(n,j)=\frac{1}{2n+1}\binom{2n+j}{j}\binom{2n+1}{n+j+1}$, for which we present a new combinatorial interpretation, connecting them to the generalized Schröder numbers of order two.

math.CO↗

arXiv2023

On new divisibility properties of generalized central trinomial coefficients and Legendre polynomials

We present a new formula for the highest power of $a+b$ that divides the sum $B(n,m,a,b)=\sum_{k=0}^{n}\binom{n}{k}^m a^{n-k}b^k$ for the case $m=2$. By using this formula, we give complete 3-adic valuation for central Dellanoy numbers. Also, we find the highest power of an odd integer $x$ that divides Legendre's polynomial $P_{n}(x)$. By using the same idea, generalized trinomial coefficients and generalized Motzkin numbers are treated. As a result, we give complete 3-adic valuation for little Schröder numbers and restricted hexagonal numbers. By using new class of binomial sums, we examine divisibility of $B(n,m, a,b)$ by powers of $a+b$ for $m >2$.

math.NT↗

arXiv2023

Is This a New Class of Matrices?

We consider a new class of matrices associated to a real square matrix $A$ and to a vector $\vec{c} \in \{-1,1\}^n$ such that $c_1=1$ by using a map $φ_{\vec{c}}$ which turns out to be a conjugation of a matrix $A$ by a signature matrix. It is shown that every such matrix is similar and congruent to a matrix $A$ and that they have same permanental polynomials. There are $2^{n-1}$ maps $φ_{\vec{c}}$ and they form an abelian group under the composition of maps isomorphic to the group $({\mathbb{Z}_2}^{n-1}, +)$. A decomposition of matrices, on a symmetric and antisymmetric matrix under a map $φ_{\vec{c}}$, is considered. Particularly, it is shown that sum of all principal minors of the order two of a matrix $A$ is equal to the sum of all principal minors of the order two of their symmetric and antisymmetric parts. It is shown that any symmetric matrix and any antisymmetric matrix under the map $φ_{\vec{c}}$ are simultaneously permutation similar to certain block matrices which have two blocks. Finally, for a fixed matrix $A$, it is proved that the number of different matrices $φ_{\vec{c}}(A)$ is $2^{n-t}$, where $t$ is the number of connected components of the graph $G$ whose adjacency matrix is $A$.

math.RA↗

arXiv2022

Factors of Alternating Convolution of the Gessel Numbers

The Gessel number $P(n,r)$ is the number of the paths in plane with $(1, 0)$ and $(0,1)$ steps from $(0,0)$ to $(n+r, n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x\geq r\}$. We show that there is a close relationship between the Gessel numbers $P(n,r)$ and the super Catalan numbers $S(n,r)$. By using new sums, we prove that an alternating convolution of the Gessel numbers $P(n,r)$ is always divisible by \frac{1}{2}$S(n,r)$.

math.CO↗

arXiv2022

A Note on the Gessel Numbers

The Gessel number $P(n,r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r,n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. In this paper, we use combinatorial arguments to derive a recurrence relation between $P(n,r)$ and $P(n-1,r+1)$. Also, we give a new proof for a well-known closed formula for $P(n,r)$. Moreover, a new combinatorial interpretation for the Gessel numbers is presented.

math.CO↗

arXiv2021

On a New Alternating Convolution Formula for the Super Catalan Numbers

We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and induction. By using a new method, we prove one interesting divisibility result with super Catalan numbers.

math.CO↗