Related papers: A combinatorial identity on Galton-Watson process

The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed…

By considering the tiling of an $N$-board (a linear array of $N$ square cells of unit width) with new types of tile that we refer to as combs, we give a combinatorial interpretation of the product of two consecutive generalized Fibonacci…

For two graphs $G_1$ and $G_2$, the size Ramsey number $\hat{r}(G_1,G_2)$ is the smallest positive integer $m$ for which there exists a graph $G$ of size $m$ such that for any red-blue edge-coloring of the graph $G$, $G$ contains either a…

We present a deterministic algorithm which, given a graph G with n vertices and an integer 1<m < n, computes in n^{O(ln m)} time the sum of weights w(S) over all m-subsets S of the set of vertices of G, where w(S)=exp{gamma t m +O(1/m)}…

For a graph $G$ and a positive integer $c$, let $M_c(G)$ be the size of a subgraph of $G$ induced by a randomly sampled subset of $c$ vertices. Second-order moments of $M_c(G)$ encode part of the structure of $G$. We use this fact, coupled…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$.…

Let $G$ be a graph and $k$ be a positive integer, and let $Kc(G, k)$ denote the number of Kempe equivalence classes for the $k$-colorings of $G$. In 2006, Mohar noted that $Kc(G, k) = 1$ if $G$ is bipartite. As a generalization, we show…

For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i…

Let $f:\mathbb{R}^k\to \mathbb{R}$ be a measurable function, and let $\{U_i\}_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables. Consider the random process $Z_i=f(U_{i},...,U_{i+k-1})$. We show that for all $\ell$, there is a…

Zeckendorf proved that every integer can be written uniquely as a sum of non-adjacent Fibonacci numbers $\{1,2,3,5,\dots\}$. This has been extended to many other recurrence relations $\{G_n\}$ (with their own notion of a legal…

An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the…

Given positive integers $k$ and $\ell$ we write $G \rightarrow (K_k,K_\ell)$ if every 2-colouring of the edges of $G$ yields a red copy of $K_k$ or a blue copy of $K_\ell$ and we denote by $R(k)$ the minimum $n$ such that $K_n\rightarrow…

Let us call a simple graph on $n\geq 2$ vertices a prime gap graph if its vertex degrees are $1$ and the first $n-1$ prime gaps. We show that such a graph exists for every large $n$, and in fact for every $n\geq 2$ if we assume the Riemann…

Let $f$ be a multiplicative function which satisfies \[ f(a^2+b^2+c^2+d^2) = f(a^2+b^2)+f(c^2+d^2) \] for positive integers $a$, $b$, $c$, and $d$. We show that $f$ is the identity function provided that $f(3)\,f(11) \ne 0$. Otherwise,…

Let $G=(V,E)$ be a simple connected graph. A matching of $G$ is a set of disjoint edges of $G$. For every $n, m\in\mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$…

A beautiful theorem of Zeckendorf states that every positive integer can be uniquely decomposed as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, where $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$. For general recurrences…

A $1$-factorization of a graph $G$ is a collection of edge-disjoint perfect matchings whose union is $E(G)$. A trivial necessary condition for $G$ to admit a $1$-factorization is that $|V(G)|$ is even and $G$ is regular; the converse is…

This paper presents both a proof method and a result. The proof method presented is particularly suitable for uniformly proving families of identities satisfied by a family of recursive sequences. To illustrate the method, we study the…

Combinatorial number system represents a non-negative natural numbers as sum of binomial coefficients. This paper presents an induction proof that there exists unique representation of every non-negative natural number $m$ as sum of $r$…