A Correspondence between Chord Diagrams and Families of 0-1 Young Diagrams

Abstract

A chord diagram is a set of chords in which no pair of chords has a common endvertex. For a chord diagram \(E\) having a crossing \(S = \{ ac, bd \}\), by the chord expansion of \(E\) with respect to \(S\), we have two chord diagrams \(E_1 = (E\setminus S) \cup \{ ab, cd \}\) and \(E_2 = (E\setminus S) \cup \{ da, bc \}\). Starting from a chord diagram \(E\), by iterating expansions, we have a binary tree \(T\) such that \(E\) is a root of \(T\) and a multiset of nonintersecting chord diagrams appear in the set of leaves of \(T\). The number of leaves, which is not depending on the choice of expansions, is called the chord expansion number of \(E\). A \(0\)-\(1\) Young diagram is a Young diagram having a value of \(0\) or \(1\) for all boxes. This paper shows that the chord expansion number of some type counts the number of \(0\)-\(1\) Young diagrams under some conditions. In particular, it is shown that the chord expansion number of an \(n\)-crossing, which corresponds to the Euler number, equals the number of \(0\)-\(1\) Young diagrams of shape \((n,n-1,\ldots,1)\) such that each column has at most one \(1\) and each row has an even number of \(1\)'s.