Enumeration of Corners in Tree-like Tableaux

In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP. Comment: 26 pages, 11 figures. This is the final version for publication

Corners in Tree-Like Tableaux

In this paper, we study tree–like tableaux, combinatorial objects which exhibit a natural tree structure and are connected to the partially asymmetric simple exclusion process (PASEP). There was a conjecture made on the total number of corners in tree–like tableaux and the total number of corners in symmetric tree–like tableaux. In this paper, we prove both conjectures. Our proofs are based off of the bijection with permutation tableaux or type–B permutation tableaux and consequently, we also prove results for these tableaux. In addition, we derive the limiting distribution of the number of occupied corners in random tree–like tableaux and random symmetric tree–like tableaux.

Symmetric Isostatic Frameworks with $\ell^1$ or $\ell^\infty$ Distance Constraints

Combinatorial characterisations of minimal rigidity are obtained for symmetric $2$-dimensional bar-joint frameworks with either $\ell^1$ or $\ell^\infty$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.

STRONG LAW OF LARGE NUMBERS FOR MARKOV CHAINS INDEXED BY SPHERICALLY SYMMETRIC TREES

In this paper, we main consider spherically symmetric tree T. First, under the condition lim supn→∞ |T(n)|/|Ln|<∞, we investigate the strong law of large numbers (SLLNs) for T-indexed Markov chains on the nth level of T. Then, combining the Stolz theorem, we obtain the SLLNs on T. Finally, we get Shannon–McMillan theorem for T-indexed Markov chains. The obtained theorems are generalizations of some known results on Cayley tree TC, N and Bethe tree TB, N.

Effects of airway tree asymmetry on the emergence and spatial persistence of ventilation defects

Asymmetry and heterogeneity in the branching of the human bronchial tree are well documented, but their effects on bronchoconstriction and ventilation distribution in asthma are unclear. In a series of seminal studies, Venegas et al. have shown that bronchoconstriction may lead to self-organized patterns of patchy ventilation in a computational model that could explain areas of poor ventilation [ventilation defects (VDefs)] observed in positron emission tomography images during induced bronchoconstriction. To investigate effects of anatomic asymmetry on the emergence of VDefs we used the symmetric tree computational model that Venegas and Winkler developed using different trees, including an anatomic human airway tree provided by M. Tawhai (University of Auckland), a symmetric tree, and three trees with intermediate asymmetry (Venegas JG, Winkler T, Musch G, Vidal Melo MF, Layfield D, Tgavalekos N, Fischman AJ, Callahan RJ, Bellani G, Harris RS. Nature 434: 777–782, 2005 and Winkler T, Venegas JG. J Appl Physiol 103: 655–663, 2007). Ventilation patterns, lung resistance (RL), lung elastance (EL), and the entropy of the ventilation distribution were compared at different levels of airway smooth muscle activation. We found VDefs emerging in both symmetric and asymmetric trees, but VDef locations were largely persistent in asymmetric trees, and bronchoconstriction reached steady state sooner than in a symmetric tree. Interestingly, bronchoconstriction in the asymmetric tree resulted in lower RL (∼%50) and greater EL (∼%25). We found that VDefs were universally caused by airway instability, but asymmetry in airway branching led to local triggers for the self-organized patchiness in ventilation and resulted in persistent locations of VDefs. These findings help to explain the emergence and the persistence in location of VDefs found in imaging studies.