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 $(n,n-1,\dots ,1)$ such that each column has at most one $1$ and each row has an even number of $1$'s.

arXiv comment11 pages, 10 figures