
Online Journal of Analytic Combinatorics
ISSN 1931-3365 (online)
The Online Journal of Analytic Combinatorics (OJAC) is a peer-reviewed electronic journal previously hosted by the University of Rochester and now published by Combinatorial Press. OJAC features research articles that span a broad spectrum of topics, including analysis, number theory, and combinatorics, with a focus on the convergence and interplay between these disciplines. The journal particularly welcomes submissions that incorporate one or more of the following elements: combinatorial results derived using analytic methods, analytic results achieved through combinatorial approaches, or a synthesis of combinatorics and analysis in either the methodologies or their applications
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- Research article
- https://doi.org/10.61091/ojac-1106
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 11, 2016
- Pages: 1-9 (Paper #6)
- Published: 31/12/2016
An involution is a permutation that is its own inverse. Given a permutation
The proof is based upon the observation that, for most permutations
- Research article
- https://doi.org/10.61091/ojac-1105
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 11, 2016
- Pages: 1-9 (Paper #5)
- Published: 31/12/2016
The Stirling number of the second kind
- Research article
- https://doi.org/10.61091/ojac-1104
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 11, 2016
- Pages: 1-9 (Paper #4)
- Published: 31/12/2016
Packing patterns in permutations concerns finding the permutation with the maximum number of a prescribed pattern. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently the packing density for all but two (up to equivalence) patterns up to length 4 can be obtained. In this note we consider the analogous question for colored patterns and permutations. By introducing the concept of “colored blocks” we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities.
- Research article
- https://doi.org/10.61091/ojac-1103
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 11, 2016
- Pages: 1-21 (Paper #3)
- Published: 31/12/2016
We extend the main result of the paper “Arithmetic progressions in sets of fractional dimension” ([12]) in two ways. Recall that in [12], Łaba and Pramanik proved that any measure
- Research article
- https://doi.org/10.61091/ojac-1102
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 11, 2016
- Pages: 1-25 (Paper #2)
- Published: 31/12/2016
Let
- Research article
- https://doi.org/10.61091/ojac-1101
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 11, 2016
- Pages: 1-17 (Paper #1)
- Published: 31/12/2016
We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group
This problem was first studied by Green [Gre05] who proved a lower bound on
The improved lower bound was already known to follow (for triangle-removal in all groups) using Fox’s removal lemma for directed cycles and a reduction by Král, Serra, and Vena~\cite{KSV09} (see [Fox11, CF13]). The purpose of this note is to provide a direct Fourier-analytic proof for the group
- Research article
- https://doi.org/10.61091/ojac-1006
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 10, 2015
- Pages: 1-18 (Paper #6)
- Published: 31/12/2015
A set of natural numbers tiles the plane if a square-tiling of the plane exists using exactly one square of side length n for every n in the set. In [9] it is shown that N, the set of all natural numbers, tiles the plane. We answer here a number of questions from that paper. We show that there is a simple tiling of the plane (no nontrivial subset of squares forms a rectangle). We show that neither the odd numbers nor the prime numbers tile the plane. We show that N can tile many, even infinitely many planes.
- Research article
- https://doi.org/10.61091/ojac-1005
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 10, 2015
- Pages: 1-15 (Paper #5)
- Published: 31/12/2015
Let
- Research article
- https://doi.org/10.61091/ojac-1004
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 10, 2015
- Pages: 1-5 (Paper #4)
- Published: 31/12/2015
In the paper, utilizing respectively the induction, a generating function of the Lah numbers, the Chu-Vandermonde summation formula, an inversion formula, the Gauss hypergeometric series, and two generating functions of Stirling numbers of the first kind, the authors collect and provide six proofs for an identity of the Lah numbers.
- Research article
- https://doi.org/10.61091/ojac-1003
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 10, 2015
- Pages: 1-9 (Paper #3)
- Published: 31/12/2015
We prove that if
where
The proof relies on a result in
where