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

Toufik Mansour 1
1Department of Mathematics, University of Haifa, Haifa 31905, Israel
Abstract:

We study the generating functions for pattern-restricted k-ary words of length n corresponding to the longest alternating subsequence statistic in which the pattern is any one of the six permutations of length three.

Paul H. Koester1
1Department of Mathematics Indiana University Bloomington, IN 47405 U. S. A.
Abstract:

We extend an argument of Felix Behrend to show that fairly dense subsets of the integers exist which contain no solution to certain systems of linear equations.

Jean-Luc Marichal 1, Michael J. Mossinghoff 2
1Institute of Mathematics University of Luxembourg 162A, avenue de la Fa¨ıencerie L-1511 Luxembourg Luxembourg
2Department of Mathematics Davidson College Davidson, NC 28035-6996 USA
Abstract:

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Polya’s Ph.D. thesis, and we discuss several applications of these formulas.

Ernie Croot1
1Georgia Institute of Technology School of Mathematics 267 Skiles Atlanta, GA 30332
Abstract:

Let F2n be the finite field of cardinality 2n. For all large n, any subset AF2n×F2n of cardinality
|A|4nloglognlogn,
must contain three points {(x,y),(x+d,y),(x,y+d)} for x,y,dF2n and d0. Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on logn, which is larger than has been obtained previously.

A.K. Agarwal1
1Centre for Advanced Study in Mathematics Panjab University Chandigarh-160014, India
Abstract:

In 1972, Bender and Knuth established a bijection between certain infinite matrices of non-negative integers and plane partitions and in [2] a bijection between Bender-Knuth matrices and n-color partitions was shown. Here we use this later bijection and translate the recently found n-color partition theoretic interpretations of four mock theta functions of S. Ramanujan in [1] to new combinatorial interpretations of the same mock theta functions involving Bender-Knuth matrices.

Victor H. Moll 1
1Department of Mathematics Tulane University New Orleans, LA 70118
Abstract:

We present analytical properties of a sequence of integers related to the evaluation of a rational integral. We also discuss an algorithm for the evaluation of the 2-adic valuation of these integers that has a combinatorial interpretation.

Matthias Schork 1
1Alexanderstr. 76 60489 Frankfurt, Germany
Abstract:

It is proposed that finding the recursion relation and generating function for the (colored) Motzkin numbers of higher rank introduced recently is an interesting problem.

Michael T. Lacey 1, William McClain 1
1School of Mathematics Georgia Institute of Technology Atlanta GA 30332
Abstract:

Let F2n be the finite field of cardinality 2n. For all large n, any subset AF2n×F2n of cardinality |A|4nloglognlogn, must contain three points {(x,y),(x+d,y),(x,y+d)} for x,y,dF2n and d0. Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on logn, which is larger than has been obtained previously.

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