John Schmitt (Middlebury College) - A combinatorial problem given to us by Dan Archdeacon 

February 3rd, 4:00 PM, Innovation E432

 

Five years ago this month Dan Archdeacon passed away, leaving us with good memories and a rich set of mathematical results, problems and conjectures.  We consider one such conjecture which arose as the result of his study of Heffter arrays, a combinatorial design that he introduced shortly before his death to facilitate the study of embedding graphs on surfaces.  We state the conjecture here.

Let (G,+) be an abelian group and consider a subset A of G with |A|=k.  Given an ordering (a1, ..., ak) of the elements of A, define its partial sums by s0 = 0 and sj=a1+a2+...+aj, for j=1,..,k.  

 Archdeacon’s Conjecture: For any cyclic group Zn and any subset A of Zn\{0}, it is possible to find an ordering of the elements of A such that no two of its partial sums si and sj are equal for i,j=1,..,k.

 We show how Noga Alon's Combinatorial Nullstellensatz can be used to frame this conjecture and, in the case that n is prime, we verify computationally that Dan’s conjecture is true for small values of |A|.  In the case that n is prime, we also show that a sequence of length k having distinct partial sums exists in any subset of Zn\{0} of size at least 2k-sqrt(8k) in all but at most a bounded number of cases.

 This is joint work with Jacob Hicks (U. Georgia) and Matt Ollis (Marlboro College).

PUBLISHED

01-28-2020
Kiki M Reno