
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
Information Menu
- Research article
- https://doi.org/10.61091/ojac-1002
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 10, 2015
- Pages: 1-12 (Paper #2)
- Published: 31/12/2015
By extending former results of Ehrhart, it was shown by Peter McMullen that the number of lattice points in the Minkowski-sum of dilated rational polytopes is a quasipolynomial function in the dilation factors. Here we take a closer look at the coefficients of these quasi-polynomials and show that they are piecewise polynomials themselves and that they are related to each other by a simple differential equation. As a corollary, we obtain a refinement of former results on lattice points in vector dilated polytopes
- Research article
- https://doi.org/10.61091/ojac-1001
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 10, 2015
- Pages: 1-11 (Paper #1)
- Published: 31/12/2015
Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Eulerian numbers
- Research article
- https://doi.org/10.61091/ojac-907
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-26 (Paper #7)
- Published: 31/12/2014
We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate
- Research article
- https://doi.org/10.61091/ojac-906
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-12 (Paper #6)
- Published: 31/12/2014
Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e., Baxter permutations
- Research article
- https://doi.org/10.61091/ojac-905
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-36 (Paper #5)
- Published: 31/12/2014
In his celebrated proof of Szemerédi’s theorem that a set of integers of positive density contains arbitrarily long arithmetic progressions, W. T. Gowers introduced a certain sequence of norms
This question has been answered fairly completely by B. Green, T. Tao and T. Ziegler in terms of certain algebraic functions called \textit{nilsequences}. In this work, we show that more explicit functions called \textit{bracket polynomials} have `large’ Gowers norm. Specifically, for a fairly large class of bracket polynomials, called \textit{constant-free bracket polynomials}, we show that if
We establish this result by first reducing it to a certain recurrence property of sets of constant-free bracket polynomials. Specifically, we show that if
The proof of this statement relies on two deep results from the literature. The first is work of V. Bergelson and A. Leibman showing that an arbitrary bracket polynomial can be expressed in terms of a so-called \textit{polynomial sequence} on a nilmanifold. The second is a theorem of B. Green and T. Tao describing the quantitative distribution properties of such polynomial sequences.
In the special cases of the bracket polynomials
- Research article
- https://doi.org/10.61091/ojac-904
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-11 (Paper #4)
- Published: 31/12/2014
The theory of generic smooth closed plane curves initiated by Vladimir Arnold is a beautiful fusion of topology, combinatorics, and analysis. The theory remains fairly undeveloped. We review existing methods to describe generic smooth closed plane curves combinatorially, introduce a new one, and give an algorithm for efficient computation of Arnold’s invariants. Our results provide a good source of future research projects that involve computer experiments with plane curves. The reader is not required to have background in topology and even undergraduate students with basic knowledge of differential geometry and graph theory will easily understand our paper.
- Research article
- https://doi.org/10.61091/ojac-903
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-18 (Paper #3)
- Published: 31/12/2014
A level (
In this paper, we consider the problem of enumerating the members of
- Research article
- https://doi.org/10.61091/ojac-902
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-17 (Paper #2)
- Published: 31/12/2014
Let
- Research article
- https://doi.org/10.61091/ojac-901
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 9, 2014
- Pages: 1-20 (Paper #1)
- Published: 31/12/2014
This statistic, i.e. the sum of positions of records, has been the object of recent interest in the literature. Using the saddle point method, we obtain from the generating function of the sum of positions of records in random permutations and Cauchy’s integral formula, asymptotic results in central and non-central regions. In the non-central region, we derive asymptotic expansions generalizing some results by Kortchemski. In the central region, we obtain a limiting distribution related to Dickman’s function. This paper fits within the framework of Analytic Combinatorics.
- Research article
- https://doi.org/10.61091/ojac-806
- Full Text
- Online Journal of Analytic Combinatorics
- Issue 8, 2013
- Pages: 1-33 (Paper #6)
- Published: 31/12/2013
We exhibit proofs of Furstenberg’s Multiple Recurrence Theorem and of a special case of Furstenberg and Katznelson’s multidimensional version of this theorem, using an analog of the density-increment argument of Roth and Gowers. The second of these results requires also an analog of some recent finitary work by Shkredov.
Many proofs of these multiple recurrence theorems are already known. However, the approach of this paper sheds some further light on the well-known heuristic correspondence between the ergodic-theoretic and combinatorial aspects of multiple recurrence and Szemeredi’s Theorem. Focusing on the density- increment strategy highlights several close points of connection between these settings.