In this paper, we prove a surprisingly simple formula that counts connected cycle-free families of set partitions, labelled free cacti and coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We also provide a formula that enumerates noncrossing connected, cycle-free pairs of partitions.
The Erdős-Anning Theorem states that an integer distance set in the Euclidean plane must have all of its points on a single line or is finite. However, this is not true if we consider area sets. That is, if \((x_1,y_1)\) and \((x_2,y_2)\) are any two vectors contained in the integer lattice, then the area of the parallelogram determined by the two vectors is an integer, showing that the points do not have to lie on a line. We prove a finite field version of these results for \(d=2\) and \(d=3\), showing that if \(E \subset \Bbb{F}_q^d, q=p^2\), where \(p\) is an odd prime and the distance set of \(E\) is \(\Bbb{F}_p\), then the size of \(E\) is at most \(p^d\). Furthermore, we prove that if the area set of \(E\) is a subset of \(\Bbb{F}_p\), then the size of \(E\) is at most \(p^2\) in two dimensions.
This paper fits in the intersection between two disparate areas of combinatorics. Namely, graph theory and the combinatorics of Catalan words. A Catalan word with n parts is defined as a word w = w1w2⋯wn over the set of positive integers in which w1 = 1 and 1 ≤ wk ≤ wk − 1 + 1 for k = 2, 3, …, n. In order to study the intersection of the two areas, a specific type of graph called a grid graph is defined for each Catalan word. The main thrust of the paper is investigating the degrees of vertices in grid graphs. For each of the possible fixed degrees i ∈ {1, 2, 3, 4}, we find generating functions DFi(x) where the coefficient of xn is the total number of vertices of degree i in all grid graphs with n parts.
Let Fn denote the n-th Fibonacci number defined by Fn = Fn − 1 + Fn − 2 if n ≥ 2, with F0 = 0 and F1 = 1. In this paper, we find determinant identities for several Toeplitz–Hessenberg matrices whose nonzero entries are derived from the sequence kn + m for various fixed m, where kn = Fn − 1. These results may be obtained algebraically as special cases of more general formulas involving the Horadam numbers and the generating functions for the associated sequences of determinants. Equivalent multi-sum identities featuring sums of products of kn terms with multinomial coefficients may be given, which follow from Trudi’s formula. Connections are made to several OEIS entries that have arisen previously in other contexts, perhaps most notably the Padovan number sequence. Finally, we provide combinatorial proofs of our identities involving kn by enumerating (or finding the sum of signs of) various classes of tilings containing squares, dominos, trominos and a special type of tile which can be of arbitrary length.
We numerically investigate typical graphs in a region of the Strauss model of random graphs with constraints on the densities of edges and triangles. This region, where typical graphs had been expected to be bipodal but turned out to be tripodal, involves edge densities e below e_0 = (3-√(3))/6 ≈ 0.2113 and triangle densities t slightly below e^3. We determine the extent of this region in (e,t) space and show that there is a discontinuous phase transition at the boundary between this region and a bipodal phase. We further show that there is at least one phase transition within this region, where the parameters describing typical graphs change discontinuously.
Let US be the class of all ultrametric spaces generated by labeled star graphs. We prove that compact US-spaces are the completions of totally bounded ultrametric spaces generated by decreasingly labeled rays. We characterize the ultrametric spaces which are weakly similar to finite US-spaces and describe these spaces by certain four-point conditions.
This note derives asymptotic upper and lower bounds for the number of planted plane trees on n nodes assigned labels from the set {1,2,…, k} with the restriction that on any path from the root to a leaf, the labels must strictly decrease. We illustrate an application to calculating the largest eigenvalue of the adjacency matrix of a tree.
The aim of this paper is to introduce and study a new class of analytic functions which generalize the classes of \(\lambda\)-Spirallike Janowski functions. In particular, we gave the representation theorem, the right side of the covering theorem, starlikeness estimates and some properties related to the functions in the class \( S_\lambda ( T, H, F ) \).
We prove two conjectures due to Sun concerning binomial-harmonic sums. First, we introduce a proof of a formula for Catalan’s constant that had been conjectured by Sun in 2014. Then, using a similar approach as in our first proof, we solve an open problem due to Sun involving the sequence of alternating odd harmonic numbers. Our methods, more broadly, allow us to reduce difficult binomial-harmonic sums to finite combinations of dilogarithms that are evaluable using previously known algorithms.
The aim of this work is to establish congruences \( \pmod{p^2} \) involving the trinomial coefficients \( \binom{np-1}{p-1}_2 \) and \( \binom{np-1}{(p-1)/2}_2 \) arising from the expansion of the powers of the polynomial \( 1 + x + x^2 \). In main results, we extend some known congruences involving the binomial coefficients \( \binom{np-1}{p-1} \) and \( \binom{np-1}{(p-1)/2} \), and establish congruences linking binomial coefficients and harmonic numbers.
Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define triangular \( q \)-binomial coefficients which are analogous to standard \( q \)-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing) partitions of \( n \).
In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-\( m \)-Pell compositions. We find that these are in bijection with certain weakly unimodal \( m \)-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects. In the special case of \( m = 2 \), we have a new combinatorial interpretation of the semi-Pell sequence and connections to other objects.
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the "complexity" of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses. In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
criteria to verify log-convexity of sequences is presented. Iterating this criteria produces infinitely log-convex sequences. As an application, several classical examples of sequences arising in Combinatorics and Special Functions are presented. The paper concludes with a conjecture regarding coefficients of chromatic polynomials.
In this paper we show some identities come from the \( q \)-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer \( n \) and the number of integer partitions of \( n \). One of the most intriguing results found here is given by the next equation, for \( n > 0 \), \[ \sum_{l=1}^n \frac{1}{l!} \sum_{w_1+w_2+\cdots+w_l \in C(n)} \frac{\sigma_1(w_1) \sigma_1(w_2) \cdots \sigma_1(w_l)}{w_1 w_2 \cdots w_l} = p_1(n), \] where \( \sigma_1(n) \) is the sum of all positive divisors of \( n \), \( p_1(n) \) is the number of integer partitions of \( n \), and \( C(n) \) is the set of integer compositions of \( n \). In the last section, we show seven applications, one of them is a series expansion for \[ \frac{(q^{a_1};q^{b_1})_\infty (q^{a_2};q^{b_2})_\infty \cdots (q^{a_k};q^{b_k})_\infty} {(q^{c_1};q^{d_1})_\infty (q^{c_2};q^{d_2})_\infty \cdots (q^{c_r};q^{d_r})_\infty}, \] where \( a_1, \ldots, a_k, b_1, \ldots, b_k, c_1, \ldots, c_r, d_1, \ldots, d_r \) are positive integers, and \( |q| < 1 \).
The notion of length spectrum for natural numbers was introduced by Pong in [5]. In this article, we answer the question of how often one can recover a random number from its length spectrum. We also include a quick deduction of a result of LeVeque in [4] on the average order of the size of length spectra.
Between Bernoulli/Euler polynomials and Pell/Lucas polynomials, convolution sums are evaluated in closed form via the generating function method. Several interesting identities involving Fibonacci and Lucas numbers are shown as consequences including those due to Byrd (1975) and Frontczak (2020).
Integer compositions and related counting problems are a rich and ubiquitous topic in enumerative combinatorics. In this paper we explore the definition of symmetric and asymmetric peaks and valleys over compositions. In particular, we compute an explicit formula for the generating function for the number of integer compositions according to the number of parts, symmetric, and asymmetric peaks and valleys.
We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these "prime-identifying" indicator functions, when we sample $m_n$ i.i.d. points uniformly from the integers $\{2, \dots, n\}$.
