Related papers: Novel structures in Stanley sequences
We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to $3 \pmod{6}$. The only previously known examples have order $15$ or $21$.
Scattered sequences are a generalization of scattered polynomials. So far, only scattered sequences of order one and two have been constructed. In this paper an infinite family of scattered sequences of order three is obtained. Equivalence…
A poset is (3+1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets are the subject of the (3+1)-free conjecture of Stanley and Stembridge. Recently, Lewis and Zhang have…
The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas…
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…
We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a…
The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…
We introduce numbers depending on three parameters which we call skyscraper numbers. We discuss properties of these numbers and their relationship with Stirling numbers of the first kind, and we also introduce a skyscraper sequence.
Several sequences of free cumulants that count binary plane trees correspond to sequences of classical cumulants that count the decreasing versions of the same trees. Using two new operations on colored binary plane trees that we call…
Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…
Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…
We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x_1,...,x_s as well as all the…
In this paper we present an extension of Stanley's theorem related to partitions of positive integers. Stanley's theorem states a relation between "the sum of the numbers of distinct members in the partitions of a positive integer $n$" and…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
We define impulse response sequence in the set of all linear recurring sequences satisfying a linear recurrence relation of order $r$. The generating function and expression of the impulse response sequence are presented. Some identities of…
The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants.…
We study the problem of generating interesting integer sequences with a combinatorial interpretation. For this we introduce a two-step approach. In the first step, we generate first-order logic sentences which define some combinatorial…
In this paper, we study the possibility of designing non-trivial random CSP models by exploiting the intrinsic connection between structures and typical-case hardness. We show that constraint consistency, a notion that has been developed to…
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of…
We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two…