Related papers: Primeness of generalized parking functions
The action of the symmetric group $S_n$ on the set $Park_n$ of parking functions of size $n$ has received a great deal of attention in algebraic combinatorics. We prove that the action of $S_n$ on $Park_n$ extends to an action of $S_{n+1}$.…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating…
The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is…
In this paper, let $\mathcal{P}_{n;\leq s;k}^l$ denote a set of $k$-flaw preference sets $(a_1,...,a_n)$ with $n$ parking spaces satisfying that $1\leq a_i\leq s$ for any $i$ and $a_1=l$ and $p_{n;\leq s;k}^l=|\mathcal{P}_{n;\leq s;k}^l|$.…
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane…
In this paper, we consider the generating functions of the complete and elementary symmetric functions and provide a new generalization of these classical symmetric functions. Some classical relationships involving the complete and…
We provide operadic interpretations for two Hopf subalgebras of the algebra of parking functions. The Catalan subalgebra is identified with the free duplicial algebra on one generator, and the Schr\"oder subalgebra is interpreted by means…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers.…
We show that the notion of parkization of a word, a variant of the classical standardization, allows to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and…
In this paper, a new formula for {\pi}^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two…
We determine all permutations in two large classes of polynomials over finite fields, where the construction of the polynomials in each class involves the denominators of a class of rational functions generalizing the classical Redei…
Simple function classes have emerged as toy problems to better understand in-context-learning in transformer-based architectures used for large language models. But previously proposed simple function classes like linear regression or…
Let A=(a_(ij)) be the generic n by n circulant matrix given by a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is…
Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gau{\ss} count the remaining ones, approximately and exactly. For polynomials in two or more…
We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression.…
We survey results about prime number races, that is, results about the relative sizes of prime counting functions $\pi_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recent work by the authors on these problems.
We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of…