Let P1, . . . ,Pn be properties of graphs. A (P1, . . . ,Pn)-partition of a graph G is a partition of the vertex set V (G) into subsets V1, . . . , Vn such that the subgraph G[Vi] induced by Vi has property Pi; i = 1, . . . , n. A graph G is said to be uniquely (P1, . . . ,Pn)-partitionable if G has exactly one (P1, . . . ,Pn)-partition. A property P is called hereditary if every subgraph of ev...

3. A lower bound To obtain a lower bound we count only those sequences of refinements which include the partition 1 .2.3 . . . d . r of n into d or d + 1 parts, d of which are of different size, where 0 < r = n-$d(d + 1) < d, so that J(2n) > d > J(2n)-3. Moreover we only count sequences in which we split off 1 from each of the d 1 parts of different size greater than 1. These d 1 steps can be m...

quercetin is a dietary flavonoid which has anti-inflammatory effects. this study aimed to evaluate the influence of quercetin on histopathological aspects and airway epithelium in  allergic airway  inflammation mice model. twenty-eight balb/c mice were randomly divided into four groups: group i (control), group ii (untreated mice with allergic airway inflammation), group iii (allergic airway in...

We call a sum a1 + a2 + · · ·+ ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ · · · ≤ ak and n = a1 + a2 + · · ·+ ak. For i = 1, 2, . . . , k, we call ai the i-th part of the sum a1 + a2 + · · ·+ ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1 + a2 + · · · + ak and b1 + b2 + · · ·+ bk strongly inter...

We give a simple combinatorial proof of three identities of Warnaar. The proofs exploit involutions due to Franklin and Schur.

Theorems in the theory of partitions are closely related to basic hypergeometric series. Some identities arising in basic hypergeometric series can be interpreted in the theory of partitions using Fpartitions. In this paper, Ramanujan’s 1ψ1 summation and the q-Gauss summation are established combinatorially.

A method for computationally efficient skeletonization of three-dimensional tubular structures is reported. The method is specifically targeting skeletonization of vascular and airway tree structures in medical images but it is general and applicable to many other skeletonization tasks. The developed approach builds on the following novel concepts and properties: fast curve-thinning algorithm t...

Combined induced pluripotent stem cells and airway-on-a-chip establish the global axis of planar cell polarity in patient-derived airway sheet.