Related papers: Combinatorial Proof of the Minimal Excludant Theor…

Let $P_r(n)$ be the set of partitions of n with non negative rth differences. Let $\lambda$ be a partition chosen uniformly at random among the set $P_r(n)$. Let $d(\lambda)$ be a positive rth difference chosen uniformly at random in…

Let $M_n$ be an $n\times n$ signed random combinatorial matrix whose rows are independent and uniformly distributed over the set of $\{-1,0,1\}$-vectors with exactly $n/2$ zero coordinates. Despite the dependence induced by the row…

The rank of a skew partition $\lambda/\mu$, denoted $rank(\lambda/\mu)$, is the smallest number $r$ such that $\lambda/\mu$ is a disjoint union of $r$ border strips. Let $s_{\lambda/\mu}(1^t)$ denote the skew Schur function…

In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course…

Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$…

We show that the shifted rank, or srank, of any partition $\lambda$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_{\lambda}$ function in terms of power sum symmetric functions. This gives…

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…

For $\Lambda$-$n$-coalescents with mutation, we analyse the size $O_n$ of the partition block of $i\in\{1,\ldots,n\}$ at the time where the first mutation appears on the tree that affects $i$ and is shared with any other…

{\it In Bargmann representation, the reggeon's field theory{\color{blue} [5]} is caracterized by the non symmetrical Gribov operator $\displaystyle{H_{\lambda',\mu,\lambda} = \lambda' A^{*^{2}}A^{2} + \mu A^{*}A + i\lambda A^{*}(A +…

The Smarandache function of a positive integer $n$, denoted by $S(n)$, is defined to be the smallest positive integer $j$ such that $n$ divides the factorial $j!$. In this note, we prove that for any fixed number $k > 1$, the inequality…

Inspired by Andrews' and Newman's work on the minimal excludant or "mex" of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.

Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\…

Let $v(k)$ be the smallest integer larger than $1$ that does not occur among the denominators in any identity of the form $$ 1=\frac1{n_1}+\cdots+\frac1{n_k}, $$ where $1 \le n_1<\cdots<n_k$ are pairwise distinct integers. In their 1980…

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le \Lambda N$ for some constant $\Lambda \ge 1$. Let $X$ be an $M\times n$ random matrix…

Lambda calculus is the basis of functional programming and higher order proof assistants. However, little is known about combinatorial properties of lambda terms, in particular, about their asymptotic distribution and random generation.…

We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} \mu(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x < \exp(1.017 \times…

Let $\delta>1$ and $\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\beta$ and $\delta$ with the following property: for any $N,n$ such that $N\ge \max(N_0,\delta n)$, any $N\times n$ random…

We provide a combinatorial description of exclusion statistics in terms of minimal difference $p$ partitions. We compute the probability distribution of the number of parts in a random minimal $p$ partition. It is shown that the bosonic…

Recent results by Andrews and Merca on the number of even parts in all partitions of n into distinct parts, a(n), were derived via generating functions. This paper extends these results to the number of parts divisible by k in all the…

Let $n$ be a natural number and $\mathcal{M}$ a set of $n \times n$-matrices over the nonnegative integers such that the joint spectral radius of $\mathcal{M}$ is at most one. We show that if the zero matrix $0$ is a product of matrices in…