An infinite antichain of planar tanglegrams

TL;DR

This paper constructs an infinite antichain of planar tanglegrams, challenging previous expectations and extending the understanding of the structure of tanglegrams and permutation patterns.

Contribution

It introduces a novel infinite antichain of planar tanglegrams, expanding the theory of tanglegram partial orders and their combinatorial properties.

Findings

Established an infinite antichain of planar tanglegrams.

Extended the analogy between tanglegrams and permutation patterns.

Provided a new construction method for antichains in combinatorial structures.

Abstract

Contrary to the expectation arising from the tanglegram Kuratowski theorem of \'E. Czabarka, L.A. Sz\'ekely and S. Wagner [SIAM J. Discrete Math. 31(3): 1732--1750, (2017)], we construct an infinite antichain of planar tanglegrams with respect to the induced subtanglegram partial order. R.E. Tarjan, R. Laver, D.A. Spielman and M. B\'ona, and possibly others, showed that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain, i.e. there exists an infinite collection of permutations, such that none of them contains another as a pattern. Our construction adds a twist to the construction of Spielman and B\'ona [Electr. J. Comb, Vol. 7. N2.]